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Maths
Geometry-2 Geometry 1.. QA.. Worksheet..

Sphere

Sphere
Sphere Facts
    It is perfectly symmetrical
    It has no edges or vertices (corners)
    It is not a polyhedron
    All points on the surface are the same distance from the center
 
Surface Area = 4 × π × r2
 
Volume = (4/3) × π × r3

Cube

Cube (Hexahedron)
Cube (Hexahedron) Facts
  It has 6 Faces
  Each face has 4 edges, and is actually a square
  It has 12 Edges
  It has 8 Vertices (corner points)
  and at each vertex 3 edges meet
 
  Surface Area = 6 × (Edge Length)2
 
  Volume = (Edge Length)3

The diagram shows a cylindrical water tank of diameter 4.2 m and height 2.5 m

How many liters of water does the tank hold when full?
Use π = 22/7
 
Use the formula
Volume = π × r2 × h with π = 22/7, r = 2.1 and h = 2.5
∴ Volume = 22/7 × 2.12 × 2.5 = 34.65 m3

Using the Unit Conversion Tool, 34.65 m3 = 34,650 liters

Therefore the tank holds 34,650 liters

Prism

Prism
A prism is a polyhedron having the same cross section all along its length.
The cross section will be a polygon (a straight-edged figure). So all sides will be flat
A cross section is the shape you get when cutting straight across an object.
Prism have :
  • 2 congruent bases
  • Rectangular side faces
  • Square Prism
    Cross section
    Number of faces = 4
    Number of bases = 2
    Cube
    Cross section
    Number of faces = 4
    Number of bases = 2
    Triangular Prism
    Cross section
    Number of faces = 3
    Number of bases = 2
    Pentagonal Prism
    Cross section
    Number of faces = 5
    Number of bases = 2

    Pyramid

    Pyramid

    A pyramid is made by connecting a base to an apex
    The base is a polygon
    The sides are triangles which meet at the top (the apex).
    It is a polyhedron.
    Pyramid have :
  • 1 base
  • triangular side faces that meet at one vertex
  • Types of pyramids
    There are many types of Pyramids, and they are named after the shape of their base.

    Triangular Pyramid:
    Pyramid:
    Base
    Triangular Pyramid Facts
    It has 4 Faces
    The 3 Side Faces are Triangles
    The Base is also a Triangle
    It has 4 Vertices (corner points)
    It has 6 Edges
    It is also a Tetrahedron
     
    Volume = 1/3 × [Base Area] × Height
     
    Surface Area (when all side faces are the same):
    = [Base Area] + 1/2 × Perimeter × [Side Length]
    Square Pyramid:
    Pyramid:
    Base
    Square Pyramid Facts
    It has 5 Faces
    The 4 Side Faces are Triangles
    The Base is a Square
    It has 5 Vertices (corner points)
    It has 8 Edges
     
    Surface Area = [Base Area] +
    1/2 × Perimeter × [Slant Length]
     
    Volume = 1/3 × [Base Area] × Height

    Pentagonal Pyramid:
    Pyramid
    Base
    Pentagonal Pyramid Facts
    It has 6 Faces
    The 5 Side Faces are Triangles
    The Base is a Pentagon
    It has 6 Vertices (corner points)
    It has 10 Edges
     
    Volume = 1/3 × [Base Area] × Height
     
    Surface Area (when all side faces are the same):
    = [Base Area] + 1/2 × Perimeter × [Side Length]

    Right vs Oblique Pyramid
    This tells you where the top (apex) of the pyramid is. If the apex is directly above the center of the base, then it is a Right Pyramid, otherwise it is an Oblique Pyramid.
    Right Pyramid:
    Oblique Pyramid:

    Regular vs Irregular Pyramid
    This tells us about the shape of the base. If the base is a regular polygon, then it is a Regular Pyramid, otherwise it is an Irregular Pyramid.
    Regular Pyramid: Because it has Regular base
    Irregular Pyramid: Because it has irregular base

    Pythagoras

    Pythagoras Theorem : In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides
     
    Example 1: Solve this triangle.
    a2 + b2 = c2
    52 + 122 = c2
    25 + 144 = c2
    169 = c2
    c2 = 169
    c = √169
    c = 13
    Example 2: Solve this triangle.
    a2 + b2 = c2
    92 + b2 = 152
    81 + b2 = 225
    Take 81 from both sides:
    b2 = 144
    b = √ 144
    b = 12
     
    Example 3 : Solve this triangle.
    a2 + b2 = c2
    72 + 242 = c2
    49 + 576 = c2
    c2 = 625
    c = √ 625
    c = 25
    Example 4 : Solve this triangle.
    a2 + b2 = c2
    a2 + 112 = 162
    a2 + 121 = 256
    Take 121 from both sides:
    a2 = 256 - 121 = 135
    a = √ 135
    Example 5 : What is the length of the diagonal of a rectangle of length 3 and width 2?
    The rectangle can be divided up into two right-angled triangles, as shown in the diagram
    We can find the length of the diagonal, d, by using Pythagoras' theorem in one triangle:
    d2 = 32 + 22 = 9 + 4 = 13
    d = √ 13
    Example 6 : What is the length of the side x?
    In this question there are two right triangles. Use Pythagoras' Theorem in each of them in turn
    c2 = 62 + 82 = 36 + 64 = 100
    So c = √ 100 = 10
    Now we know the value of c, mark it in on the second triangle and use Pythagoras' Theorem again
    122 = 102 + x2
    144 = 100 + x2
    x2 = 144 - 100 = 44
    x = √ 44 = 2 √ 11
    Example 7 : The diagram shows a kite ABCD.
    The diagonals cut at right angles and intersect at O.
    What is the length of the diagonal AC?
    Use Pythagoras' Theorem twice:
    In triangle AOD
    102 = x2 + 82
    100 = x2 + 64
    x2 = 100 - 64 = 36
    x = √ 36 = 6
     
    In triangle COD
    172 = y2 + 82
    289 = y2 + 64
    y2 = 289 - 64 = 225
    y = √ 225 = 15
     
    Therefore the length of AC = x + y = 6 + 15 = 21
    Example 8 : Town B is 8 miles north and 17 miles east of town A. How far are the two towns apart?
     
    Complete the right-angled triangle ABC showing that B is 8 miles north and 17 miles east of A
    We are asked to find the distance from A to B = c miles.
    By Pythagoras:
    c2 = a2 + b2
    c2 = 82 + 172 = 64 + 289 = 353
    c = √ 353 = 18.8 correct to one decimal place.
    The two towns are 18.8 miles apart.
     
    Example 9 : A 3m ladder stands on horizontal ground and reaches 2.8 m up a vertical wall. How far is the foot of the ladder from the base of the wall?
     
    Complete a right-angled triangle ABC showing that the ladder is 3 m long and the distance up the wall is 2.8 m:
    We are asked to find the distance from the foot of the ladder to the base of the wall = "a" m.
    By Pythagoras: c2 = a2 + b2
    3 2 = a 2 + 2.8 2
    9 = a2 + 7.84
    a2 = 9 - 7.84 = 1.16
    a = √ 1.16 = 1.08 correct to 2 decimal places
     
    The foot of the ladder is 1.08 m from the base of the wall
    Example 10 : A rectangular field is 125 yards long and the length of one diagonal of the field is 150 yards. What is the width of the field?
     
    The rectangular field is two right-angled triangles (one of which is triangle ABC):
    We are asked to find the width of the field = b yds.
    By Pythagoras: c2 = a2 + b2
    1502 = 1252 + b2
    22,500 = 15,625 + b2
    b2 = 22,500 - 15,625 = 6,875
    b = v6,875 = 82.9 correct to 1 decimal place
     
    The width of the field is 82.9 yards.

    Symmetry

    Line of Symmetry
    Definition: A line that divides a figure into two congruent parts each of which is the mirror image of the other.
    These figures are symmetrical in relation to the dashed line. The line is called a symmetry line or line of symmetry.
    If you fold this figure along the line of symmetry both halves would match exactly
    Line of symmetry also called Reflection Symmetry or Mirror Symmetry.
    Equilateral Triangle
    (all sides equal, all angles equal)
    3 Lines of Symmetry
    Isosceles Triangle
    (two sides equal,vtwo angles equal)
    1 Line of Symmetry
    Scalene Triangle
    (no sides equal, no angles equal)
    No Lines of Symmetry
    A rhombus has just 2 lines of symmetry, l and m
    A rectangle have 2 lines of symmetry
    A parallelogram that has no lines of symmetry
    A star has 4 lines of symmetry
    A octagon has 4 lines of symmetry

    Area

    Area of Polygons
    1) Rectangle
    The area A of any rectangle is equal to the product of the length l and the width w.
    Formula: A = lw
    2) Square
    The area A of any square is equal to the square of the length s of a side
    Formula: A = s2
    3) Triangle
    The area A of any triangle is equal to one-half the product of any base b and corresponding height h
    Formula: A = .5bh
    4) Parallelogram
    The area A of any parallelogram is equal to the product of any base b and the corresponding height h.
    Formula: A = bh
    5) Rhombus
    The area A of any rhombus is equal to one-half the product of the lengths d1 and d2 of its diagonals
    Formula: A = .5d1d2
    6) Trapezoid
    The area A of any trapezoid is equal to one-half the product of the height h and the sum of the bases, b1 and b2.
    Formula: A = .5h(b1 + b2)
    7) Circle
    The area A of any circle is equal to the product of PI and the square of the radius r.
    Formula: A = (PI)r 2
    8) Circle : Sector Area Theorem
    The area A of any sector with an arc that has degree measure n and with radius r is equal to the product of the arc's measure divided by 360 multiplied by PI times the square of the radius.
    Formula: A = (n/360)((PI)r2)

    Trigonometry

    Trigonometry
    Trigonometry is all about triangles

      Right Angled Triangle
      A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side:
     
      Adjacent is adjacent to the angle "θ",
      Opposite is opposite the angle, and
      the longest side is the Hypotenuse.

    Angles (such as the angle "θ" above) can be in Degrees or Radians. Here are some examples:
    Angle Degrees Radians
    right angle Right Angle 90° π/2
    __ Straight Angle 180° π
    right angle Full Rotation 360° 2π

    Sine, Cosine and Tangent
    The three most common functions in trigonometry are Sine, Cosine and Tangent.
     
    A right triangle consists of one angle of 90o and two acute angles. Each acute angle of a right triangle has the properties of sine, cosine and tangent.
     
    The sine, cosine and tangent of an acute angle of a right triangle are ratios of two of the three sides of the right triangle.
     
      They are simply one side of a triangle divided by another.
    Sine Function:
    sin(θ) = Opposite / Hypotenuse
    Cosine Function:
    cos(θ) = Adjacent / Hypotenuse
    Tangent Function:
    tan(θ) = Opposite / Adjacent

      Example: What is the sine of 35°?

    Using this triangle (lengths are only to one decimal place):

    sin(35°) = Opposite / Hypotenuse = 2.8/4.9 = 0.57...

      Other Functions (Cotangent, Secant, Cosecant)
      Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:
    Cosecant Function:
    csc(θ) = Hypotenuse / Opposite
    Secant Function:
    sec(θ) = Hypotenuse / Adjacent
    Cotangent Function:
    cot(θ) = Adjacent / Opposite

      Example:
      Find the Missing Angle "C"
      It's easy to find angle C by using angles of a triangle add to 180° :
      So C = 180° - 76° - 34° = 70°

      Repeating Pattern
      Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation.
     
      When you need to calculate the function for an angle larger than a full rotation of 2π (360°) just subtract as many full rotations as you need to bring it back below 2π (360°):
     

    Example: what is the cosine of 370°?

    370° is greater than 360° so let us subtract 360°

    370° - 360° = 10°

    cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

    Glossary

    Glossary

    xx: http://www.learner.org/courses/learningmath/geometry/keyterms.html#o

    acute triangle: An acute triangle is a triangle with all three angles less than 90°.
    altitude: An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.
    angle-angle-angle (AAA) similarity: The angle-angle-angle (AAA) similarity test says that if two triangles have corresponding angles that are congruent, then the triangles are similar. Because the sum of the angles in a triangle must be 180°, we really only need to know that two pairs of corresponding angles are congruent to know the triangles are similar.
    angle bisector: An angle bisector is a ray that cuts the angle exactly in half, making two equal angles.

    central angle: A central angle is an angle with its vertex at the center of a circle.
    centroid: The centroid of a triangle is the point where the three medians meet. This point is the center of mass for the triangle. If you cut a triangle out of a piece of paper and put your pencil point at the centroid, you could balance the triangle.
    circle: A circle is the set of all points in a plane that are equidistant from a given point in the plane, which is the center of the circle.
    circumcenter: The circumcenter of a triangle is the point where the three perpendicular bisectors meet. This point is the same distance from each of the three vertices of the triangles.
    concave polygon: A concave polygon is any polygon with an angle measuring more than 180°. Concave polygons look like they are collapsed or have one or more angles dented in.
    concurrent: When three or more lines meet at a single point, they are said to be concurrent. In a triangle, the three medians, three perpendicular bisectors, three angle bisectors, and three altitudes are each concurrent.
    congruent: Two figures are congruent if all corresponding lengths are the same, and if all corresponding angles have the same measure. Colloquially, we say they "are the same size and shape," though they may have different orientation. (One might be rotated or flipped compared to the other.)
    congruent triangles: Congruent triangles are triangles that have the same size and shape. In particular, corresponding angles have the same measure, and corresponding sides have the same length.
    converse: Converse means the "if" and "then" parts of a sentence are switched. For example, "If two numbers are both even, then their sum is even" is a true statement. The converse would be "If the sum of two numbers is even, then the numbers are even," which is not a true statement.
    convex polygon: A convex polygon is any polygon that is not concave.
    coordinates: Points are geometric objects that have only location. To describe their location, we use coordinates. We begin with a standard reference frame (typically the x- and y-axes). The coordinates of a point describe where it is located with respect to this reference frame. They are given in the form (x,y) where the x represents how far the point is from 0 along the x-axis, and the y represents how far it is from 0 along the y-axis. The form (x,y) is a standard convention that allows everyone to mean the same thing when they reference any point.
    cosine: If angle A is an acute angle in a right triangle, the cosine of A is the length of the side adjacent to angle A, divided by the length of the hypotenuse of the triangle. We often abbreviate this as cos A = (adjacent)/(hypotenuse).
    cross section: A cross section is the face you get when you make one slice through an object.

    diameter: A circle's diameter is a segment that passes through the center and has its endpoints on the circle.

    edge: An edge is a line segment where two faces intersect.
    equilateral triangle: An equilateral triangle is a triangle with three equal sides.

    face: A face is a polygon by which a solid object is bound. For example, a cube has six faces. Each face is a square.
    frieze pattern: A frieze pattern is an infinite strip containing a symmetric pattern

    glide reflection: A glide reflection is a combination of two transformations: a reflection over a line followed by a translation in the same direction as the line.

    hypotenuse: The hypotenuse in a right triangle is the side of the triangle that is opposite to the right angle.

    incenter: The incenter of a triangle is the point where the three angle bisectors meet. This point is the same distance from each of the three sides of the triangle.
    inscribed angle: An inscribed angle is an angle whose vertex is on a circle and whose rays intersect the circle.
    intercept: An intercept is an intersection of a graph with one of the axes. An intersection with the horizontal axis is often referred to as an x-intercept, and an intersection with the vertical axis is often referred to as a y-intercept.
    irregular polygon: An irregular polygon is any polygon that is not regular.
    isosceles trapezoid: An isosceles trapezoid is a quadrilateral with one pair of parallel sides and congruent base angles, or it is a trapezoid with congruent base angles.
    isosceles triangle: An isosceles triangle is a triangle with two equal sides.

    kite: A kite is a quadrilateral that has two pairs of adjacent sides congruent (the same length).

    line: A line has only one dimension: length. It continues forever in two directions (so it has infinite length), but it has no width at all. A line connects two points via the shortest path, and then continues on in both directions.
    line segment: A line segment is the portion of a line lying strictly between two points. It has a finite length and no width.
    line symmetry or reflection symmetry: A polygon has line symmetry, or reflection symmetry, if you can fold it in half along a line so that the two halves match exactly. The folding line is called the line of symmetry.

    median: A median is a segment connecting any vertex of a triangle to the midpoint of the opposite side.
    midline: A midline is a segment connecting two consecutive midpoints of a triangle.
    midline theorem: The midline theorem states that a midline of a triangle creates a segment that is parallel to the base and half as long.

    net: A net is a two-dimensional representation of a three-dimensional object.

    obtuse triangle: An obtuse triangle is a triangle with one angle more than 90°.
    orthocenter: The orthocenter of a triangle is the point where the three altitudes meet, making them concurrent.

    parallel lines: Parallel lines are two lines in the same plane that never intersect. Another way to think about parallel lines is that they are "everywhere equidistant." No matter where you measure, the perpendicular distance between two parallel lines is constant.
    parallelogram: A parallelogram is a quadrilateral that has two pairs of opposite sides that are parallel.
    perpendicular bisector: The perpendicular bisector of a line segment is perpendicular to that segment and bisects it; that is, it goes through the midpoint of the segment, creating two equal segments.
    plane: A plane is a flat, two-dimensional object. We often represent a plane by a piece of paper, a blackboard, or the top of a desk. In fact, none of these is actually a plane, because a plane must continue infinitely in all directions and have no thickness at all. A plane can be defined by two intersecting lines or by three non-collinear points.
    Platonic solid: A Platonic solid is a solid such that all of its faces are congruent regular polygons and the same number of regular polygons meet at each vertex.
    point: A point specifies only location; it has no length, width, or depth. We usually represent a point with a dot on paper, but the dot we make has some dimension, while a true point has dimension 0.
    polygon: A polygon is a two-dimensional geometric figure with these characteristics: It is made of straight line segments. Each segment touches exactly two other segments, one at each of its endpoints. It is closed -- it divides the plane into two distinct regions, one inside and the other outside the polygon.
    Polyhedron: A polyhedron is a closed three-dimensional figure. All of the faces are made up of polygons.
    Pythagorean theorem: The Pythagorean theorem states that if you have a right triangle, then the square built on the hypotenuse is equal to the sum of the squares built on the other two sides. a2 + b2 = c2.

    quadrilateral: A quadrilateral is a polygon with exactly four sides.

    radius: The radius of a circle is the distance from the circle's center to a point on the circle, and is constant for a given circle.
    ray: A ray can be thought of as a half a line. It has a point on one end, and it extends infinitely in the other direction.
    rectangle: A rectangle is a quadrilateral with four right angles.
    reflection: Reflection is a rigid motion, meaning an object changes its position but not its size or shape. In a reflection, you create a mirror image of the object. There is a particular line that acts like the mirror. In reflection, the object changes its orientation (top and bottom, left and right). Depending on the location of the mirror line, the object may also change location.
    regular polygon: A regular polygon has sides that are all the same length and angles that are all the same size.
    rhombus: A rhombus is a quadrilateral that has all four sides congruent.
    right triangle: A right triangle is a triangle with one right (90°) angle.
    rotation: Rotation is a rigid motion, meaning an object changes its position but not its size or shape. In a rotation, an object is turned about a "center" point, through a particular angle. (Note that the "center" of rotation is not necessarily the "center" of the object or even a point on the object.) In a rotation, the object changes its orientation (top and bottom). Depending on the location of the center of rotation, the object may also change location.
    rotation symmetry: A figure has rotation symmetry if you can rotate (or turn) that figure around a center point by fewer than 360° and the figure appears unchanged.

    scalene triangle: A scalene triangle is a triangle with all three sides unequal.
    side-angle-side (SAS) congruence: Side-angle-side (SAS) congruence states that if any two sides of a triangle are equal in length to two sides of another triangle and the angles bewteen each pair of sides have the same measure, then the two triangles are congruent; that is, they have exactly the same shape and size.
    side-angle-side (SAS) similarity: The side-angle-side (SAS) similarity test says that if two triangles have two pairs of sides that are proportional and the included angles are congruent, then the triangles are similar.
    side-side-side (SSS) congruence: The side-side-side (SSS) congruence states that if the three sides of one triangle have the same lengths as the three sides of another triangle, then the two triangles are congruent.
    side-side-side (SSS) similarity: The side-side-side (SSS) similarity test says that if two triangles have all three pairs of sides in proportion, the triangles must be similar.
    similar: Two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion.
    similar triangles: Similar triangles are triangles that have the same shape but possibly different size. In particular, corresponding angles are congruent, and corresponding sides are in proportion.
    sine: If angle A is an acute angle in a right triangle, the sine of A is the length of the side opposite to angle A divided by the length of the hypotenuse of the triangle. We often abbreviate this as sin A = (opposite)/(hypotenuse).
    square: A square is a regular quadrilateral.
    symmetry A design has symmetry if you can move the entire design by either rotation, reflection, or translation, and the design appears unchanged.

    tangent If angle A is an acute angle in a right triangle, the tangent of A is the length of the side opposite to angle A divided by the length of the side adjacent to angle A. We often abbreviate this as tan A = (opposite)/(adjacent).
    tangram A tangram is a seven-piece puzzle made from a square. A typical tangram set contains two large isosceles right triangles, one medium isosceles right triangle, two small isosceles right triangles, a square, and a parallelogram.
    theorem A theorem in mathematics is a proven fact. A theorem about right triangles must be true for every right triangle; there can be no exceptions. Just showing that an idea works in several cases is not enough to make an idea into a theorem.
    translation Translation is a rigid motion, meaning an object changes its position but not its size or shape. In a translation, an object is moved in a given direction for a particular distance. A translation is therefore usually described by a vector, pointing in the direction of movement and with the appropriate length. In translation, the object changes its location, but not its orientation (top and bottom, left and right).
    translation symmetry Translation symmetry can be found only on an infinite strip. For translation symmetry, you can slide the whole strip some distance, and the pattern will land back on itself.
    transversal A transversal is a line that passes through (transverses) two other lines. We often consider what happens when the two other lines are parallel to each other.
    trapezoid A trapezoid is a quadrilateral that has one pair of opposite sides that are parallel.
    triangle inequality The triangle inequality says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the third length.

    van Hiele levels Van Hiele levels make up a theory of five levels of geometric thought developed by Dutch educators Pierre van Hiele and Dina van Hiele-Geldof. The levels are (0) visualization, (1) analysis, (2) informal deduction, (3) deduction, and (4) rigor. The theory is useful for thinking about what activities are appropriate for students, what activities prepare them to move to the next level, and how to design activities for students who may be at different levels.
    vector A vector can be used to describe a translation. It is drawn as an arrow. The arrowhead points in the direction of the translation, and the length of the vector tells you the length of the translation.
    Venn diagram A Venn diagram uses circles to represent relationships among sets of objects
    vertex A vertex is the point where two sides of a polygon meet.

    Practice

    Q&A

    Question 1: your rectangular ad is 5 inches long its width is 2 inches shorter than the lenth what is the area of the ad
     
      We know that to find the area of the rectangle, you use . But this is a word problem, we don't have measurements put out for us that easily. Let's write this in a mathematical notation. Your rectangular ad is 5 inches long (l=5). Its width is 2 inches shorter than the length (l-2). Since we know that l = 5, we can plug that in to get the width. . So, the length is 5 and the width is 3. The area of the rectangle is 15. :)
     
    http://www.onlinemathlearning.com/sat-practice-questions-geometry-1.html

    Question 2: What is the area of the following square, if the length of BD is ?
     
     
    Solution:

    square
    Step 1: We need to find the length of the side of the square in order to get the area.
    Step 2: The diagonal BD makes two 45°-45°-90° triangles with the sides of
        the square.
    Step 3: Using the 45°-45°-90° special triangle ratio n n n rt2. If the hypotenuse
        is2 root 2 then the legs must be 2. So the length of the side of the square is 2.
    Step 4: Area of square = s2 = 22 = 4

    Answer: (D) 4

    Question 3: In the figure below, what is the value of y ?
     
     
    Solution:

    triangles
    Step 1: Vertical angles being equal allows us to fill in two angles in the triangle that
        y° belongs to.
    Step 2: Sum of angles in a triangle = 180°.
        So, y° + 40° + 80° = 180°
            y° + 120° = 180°
            y° = 60°

    Answer: (C) 60

    Question 4: Two circles both of radii 6 have exactly one point in common. If A is a point on one circle and B is a point on the other circle, what is the maximum possible length for the line segment AB?
     
     
    Solution:

    circles
    Step 1: Sketch the two circles touching at one point.
    Step 2: The furthest that A and B can be would be at the two ends as shown in the
        above diagram.
    Step 3: If the radius is 6 then the diameter is 2 × 6 = 12 and the distance from A to
        B would be 2 × 12 = 24

    Answer: (E) 24

    Question 5: A right circular cylinder has a radius of 3 and a height of 5. Which of the following dimensions of a rectangular solid will have a volume closest to the cylinder.
     
    Solution:

    Step 1: Write down formula for volume of cylinder
                 V = πr2h
    Step 2: Plug in the values
                 V = π × 32 × 5 = 45π
                V =
    45 × 3.142 = 141.39

    Step 3: We now have to test the volume of each of the rectangular solids to find
        out which is the closest to 141.39
                 (A) 4 × 5 × 5 = 100
                 (B) 4 × 5 × 6 = 120
                 (C) 5 × 5 × 5 = 125
                 (D) 5 × 5 × 6 = 150
                 (E) 5 × 6 × 6 = 180

    Answer: (D) 5, 5, 6

    Question 6: In the figures below, x = 60. How much more is the perimeter of triangle ABC compared with the triangle DEF.
     
     
    Solution:
    triangle triangle
    Note: Figures not drawn to scale

    Step 1: Since x = 60°, triangle ABC is an equilateral triangle with sides all equal.
        The sides are all equal to 8.
        Perimeter of triangle ABC = 8 + 8 + 8 = 24.
    Step 2: Triangle DEF has two angles equal, so it must be an isosceles triangle.
        The two equal sides will be opposite the equal angles.
        So, the length of DF = length of DE = 10.
        Perimeter of triangle DEF = 10 + 10 + 4 = 24.
    Step 3: Subtract the two perimeters.
                24 – 24 = 0

    Answer to Question 5: (A) 0

    Question 7: John has a square piece of paper with sides 4 inches each. He rolled up the paper to form a cylinder. What is the volume of the cylinder?
     
    Solution:

    Step 1: The edge of the paper will form the circumference of the base of
        the cylinder.
        eqn
    Step 2: Volume of cylinder = area of base × height
        eqn
        Height = edge of the paper = 4
         eqn

    Answer: (B) 16/pi

    Question 8: The square ABCD touches the circle at 4 points. The length of the side of the square is 2 cm. Find the area of the shaded region.
     
     
    Solution:

             diagram
    Step 1: Area of shaded region = Area of circle – area of square.
        We need to get the area of the circle and area of the square.
    Step 2: The diagonal BD makes two 45°-45°-90° triangles with the sides of
        the square.
    Step 3: Using the 45°-45°-90° special triangle ratio n:n:n root 2. If the leg is 2
        then the diagonal BD must be 2 rt 2.
    Step 4: BD is also the diameter of the circle.
        radius
        area of circle
    Step 5: Area of square = 2 × 2 = 4
    Step 6: Area of shaded region = Area of circle – area of square = 2π – 4

    Answer: (B) 2π – 4

    Question 9: In the figure below, if x < 90° then which of the following must be true?
     
     
    Solution:

    Step 1: x and y are supplementary angles. So if x < 90° then y > 90°
         Beware! Although the figure seems to show that l is parallel to m, it is not
        stated in the question.

    Answer: (E) y > 90°

    Question 10: In the figure below, and and AE = EF . What is the value of x?
     
     
    Solution:


    Step 1: We can find the supplementary angle of 120°, which gives us 60°.

    Step 2: is parallel to because they are both perpendicular to . This means that since they are corresponding angles.

    Step 3: Triangle ABE and triangle ACF are similar triangles , since and (AA rule). Given that AE = EF, we can conclude that AB = BC

    Step 4: Triangle ABE and triangle CBE are congruent triangles (SAS rule). So, x = angle AEB = 60°

    So, x = 60


    Answer: (E) 60

    Question 11 : In the figure below, if ∠AOB = 40° and the length of arc AB is 4π, what is the area of the sector AOB?
     
     
    Solution: Step 1: Arc AB = 40°. Circumference of circle = 360°
        Let C = circumference of circle
        equation
    Step 2: You are given that arc AB = 4π. Plug into above equation.
        equation
    Step 3: Using the formula for the circumference of circle: C = 2πr. Plug into the above equation.
        2πr = 36π
        2r = 36
        r = 18
    Step 4: Using the formula for the area of circle: A = πr2. Plug in value for r.
        A = π(18)2 = 324π
    Step 5: Sector AOB is 1/9 of the area of the circle.
        Sector AOB = 1/9 × 324π = 36π

    Answer: (C) 36π