Math 3

​Geometry

By Joanna Li

Lines & Angles

• Ray: Straight line that begins at an endpoint and extends infinitely in one direction

• Line segment: Consists of all points between 2 endpoints

• Midpoint: Point exactly halfway between 2 other points

  • Divides segment into 2 equal segments

  • Two line segments of equal length are congruent

Lines & Angles

Multiple Choice

WXYZ is a shape that has 4 sides. Side XY is perpendicular to side WX and side YZ. How many lines in WXYZ must be parallel?

• 0

• 1

• 2

• 4

Angles

• Supplementary Angles: Angles that sum to 180 degrees

  • Angles in a straight line always sum to 180 degrees

• Sum of angles around a point equal to 360 degrees

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• A transversal across parallel lines creates congruent angles

  • Big/Small Rule: All acute angles are equal, and all obtuse angles are equal

Multiple Choice

Lines A and B are parallel lines cut by a transversal. Find the value of x.

• x = 8

• x = 13

• x = 74

• x = 106

Angles

• Vertical Angles: When lines intersect, the angles that share a vertex and are across from each other are called vertical angles and are equal in measure

Polygons

Multiple Choice

What is the sum of a Quadrilaterals interior angles?

• 180
• 360
• 400
• 240

Multiple Choice

What is the value of x?

• 4
• 360
• 12
• 6

Area & Perimeter

• Area: Number of square units needed to cover a shape

• Perimeter: Sum of all side lengths

Congruence/Similarity

• Congruent Polygons have identical size & shape

• Similar Polygons have equal angles and proportional sides

Congruence/Similarity

• Polygon similarity may be used to test your grasp on proportionality

• Proportion: A statement that 2 ratios are equal

  • If you have 2 apples for every orange, it's also true that you have 4 apples for every 2 oranges

• When two ratios are equal, so are their cross products

Triangles

• Triangle Inequality Theorem: Sum of any 2 sides (a,b) must be greater than the third side (c), and the absolute difference of any 2 sides (a,b) must be less than the third side (c).

Triangles

Classifying Triangles by Side Length

• Equilateral: All 3 sides are congruent

  • All angles are 60 degrees

• Isoceles: Exactly 2 sides are congruent

  • The base angles will be congruent

• Scalene: No sides are congruent

  • No angles will be congruent

Classifying Triangles by Angles

• Equiangular: All 3 angles are congruent

  • All 3 sides will be congruent

• Right: Exactly 1 angle equals 90 degrees

  • Pythagorean theorem

• Acute: All 3 angles are less than 90 degrees

• Obtuse: Exactly 1 angles is greater than 90 degrees

Pythagorean Theorem

• Allows you to find a side length or hypoteneuse for a right triangle given the other 2 sides

Pythagorean Triples

• Pythagorean Triple: All 3 sides of a right triangles are integers

• A few common ones tested on the ACT are:

  • 3:4:5

  • 5:12:13

  • 7:24:25

• Be careful -- a 3:4 ratio doesn't always mean the third leg is 5

  • Here, you must make sure you're dealing with a right triangle and solving for the hypotenuse

Multiple Choice

In the figure shown, $$AB\parallel DE$$ , C is the intersection of AE and BD, $$\angle D$$ is a right angle, and the given side lengths are in inches. What is the length of AC, in inches?

• 4

• 15/4

• 25/4

• 25/3

Special Right Triangles

• 45-45-90: Right isosceles triangle

• 30-60-90: 30° < 60° < 90°, so they're opposite to the sides x < x√3 < 2x

Multiple Choice

A flagpole casts a shadow on the ground. The angle of elevation from the tip of the shadow to the top of the flagpole is $$60^{\circ}$$ . If the distance from the base of the pole to the tip of the shadow is 6 feet, how tall is the flagpole?

• 6

• $$6\sqrt{3}$$

• 12

• $$12\sqrt{3}$$

Multiple Choice

$$\triangle ABC$$ is an equilateral triangle with a side length of 8. What is the area of the triangle?

• 16

• $$16\sqrt[]{3}$$

• 32

• $$32\sqrt[]{3}$$

Multiple Choice

A square garden has a diagonal path that is $$10\sqrt{2}$$ feet long. What is the area of the garden, in square feet?

• 10

• 20

• 100

• 200

Quadrilaterals

Quadrilaterals

• Square: A rectangle with 4 equal sides

  • You only need one value -- the diagonal, area, or perimeter -- you can calculate all other info

Multiple Choice

The midpoints of the sides of square ABCD are the vertices of square KLMN. The side lengths of square ABCD is 14 m. What is the area, in square meters, of the shaded portion?

• 49

• $$14\sqrt[]{2}$$

• $$7\sqrt[]{2}$$

• 98

Multiple Choice

A square and a rectangle have the same area. The length of the rectangle is 196 cm, and the width of the rectangle is 4 cm. What is the length, in cm, of a side of the square?

• 20

• 28

• 100

• 400

Parallelograms

• Paralellogram: Quadrilateral formed by 2 parallel lines cut by two parallel transversals

  • Big/Small rule states obtuse angles are equal and acute angles are equal

Parallelograms

• Rhombus: Parallelogram with 4 equal sides

  • Use standard area/perimeter formulas for a parallelogram

Circles

• Circle: Set of points in a plane that are a fixed distance from a given point, called the center

• Circumerence: Perimeter of a circle

Multiple Choice

What is the AREA in square feet, of the smallest square that can contain a circle with a radius of 3 feet?

the image represents the circle inside of the square

• 36

• 6

• 12

• 9

Circles

• Formula of a circle is

where (h,k) is the center of the circle and r is the radius

Sectors and Arcs

• Sector: Area of a slice of the circle

Multiple Choice

The area of the shaded sector is $$6\pi$$ . If the central angle is $$60^{\circ}$$ , what is the radius of the circle?

• 3

• 6

• 12

• 36

Sectors and Arcs

• Arc: Segment of the circumference of a circle

Multiple Choice

A circle has a radius of 9 cm. What is the length, in centimeters, of an arc intercepted by a central angle of $$40^{\circ}$$ ?

• $$\pi$$

• $$2\pi$$

• $$4\pi$$

• $$18\pi$$

Solid Geometry

• Surface Area: Total area of all sides in a 3D object

• Volume: Amount of space within an object, measured in units³

Multiple Choice

A solid rectangular block of steel weighs w pounds. Which of the following expressions gives the weight, in pounds, of a larger solid rectangular block that is made of the same kind of steel and has length, width, and height 4 times of those of the smaller block? Assume weight is proportional to volume.

• 4w

• 16w

• 64w

• 64w³

Distance Formula

Multiple Choice

What is the distance, in coordinate units, between the points (−3, 2) and (1, 5)?

• 4

• 5

• $$\sqrt[]{7}$$

• $$\sqrt[]{13}$$

Multiple Choice

A circle in the coordinate plane has a center at (2, -3) and passes through the point (5, 1). What is the length of the diameter of the circle?

• 5

• 10

• 25

• $$5\pi$$

Midpoint Formula

• You find the averages of x- and y-coordinates of the endpoints

Multiple Choice

In the standard (x, y) coordinate plane, what is the midpoint of the line segment with endpoints (−4, 8) and (6, 2)?

• (1,5)

• (2,10)

• (5,3)

• (-5,5)

Multiple Choice

The midpoint of segment AB is (3, −1). If the coordinates of point A are (−1, 5), what are the coordinates of point B?

• (1,2)

• (2,4)

• (7,-7)

• (7,4)

Geometric Transformations

• Changes an object's position or oreintation without affecting sape and size

  • Can help evaluate congruence and symmetry

• Translation: Moving an object without rotation or reflection

• Rotation: Turning an object around a point

• Reflection: Producing a mirror image of an object across a line of reflection

Geometric Transformations

• Symmetry: Correspondence in size, shape, and relative position of parts on opposite sides of a dividing line or median plane or about a center of axis.

  • Shapes may be symmetrical about one axis, but not another

Coordinate Geometry