Geometry

Question 1

Triangle Properties

Answer 1

The exterior angle is created by one side of a triangle and the extension of the adjacent side.

The sum of all exterior angles (count 1/vertex) in a polygon is always 360. Each vertex has 2 exterior angles.

Use interior angle sums to 180 to set-up expressions with variables.

An exterior angle is equal to the sum of its two remote interior angles

The sum of the lengths of any two sides is greater than the length of the third side. The difference of lengths of any two sides is less than the length of the third side.

Circle Inscribed in Equilateral Triange:

• It is the largest circle that can fit in the triangle.
• It will split each side of the triangle into halves
• 30-60-90 triangles can be formed from the radius of the circle.

Question 2

Triangle Types

Answer 2

Pythagorean Triples Ratios: 3:4:5 and 5:12:13

Isosceles Triangle (45-45-90):
Two triangles add to a square
Area: s^2/2
Sides: x:x:x*root(2)

Scalene Triangle (30-60-90)
x:x*root(3):2x

Equilateral Triangle:
Area: s^2*root(3)/4

Question 3

Similar Triangles

Answer 3

Similar Triangle Determination:

1. All angles are the same (as long as 2 are the same. the third will be the same also)
2. All sides have the same ratio
3. One vertex angle is the same, and the corresponding sides are in the same ratio.

Question 4

Proportional Figures

Answer 4

In Problem Solving Questions, the diagrams will be drawn to scale. Use this to set-up ratios with similar triangles.

Question 5

Quadrilaterals

Answer 5

Parallelogram

• Opposite sides are equal in length
• Opposite angles are equal in measure
• Diagonals bisect each other
• Each diagonal bisects the parallelogram into two congruent triangles
• Consecutive angles add-up to 180
• Area = b*h

Question 6

Rectangles

Answer 6

Properties

1. A rectangle has 2 diagonals
2. The diagonal is sqrt(length^2+width^2) - It’s the hypotenuse of the two right triangles formed
3. The diagonals are of the same length and bisect each other

Question 7

Squares

Answer 7

Properties
All sides equal length
Diagonals are perpendicular to each other
Diagonal bisect each other and create 45-45-90 triangles
Length of Diagonal = s*sqrt(2)

Give the same perimeter, a square will always have greater area than a rectangle. A circle always has a greater area than a square, given same perimeter

Given the same area, a square will always have the least perimeter

Square inscribed in square:
The area of an inscribed square will be the smallest when the vertices are located at the midpoint of the respective edge. At this point, the square will have half the area of the larger square.

Circles inscribed in squares:
When a circle is inscribed in a square, the diameter has the same length as one of the sides

Triangle inscribed in a square:
The base and height of the triangle will be the same length as the side of the square

Question 8

Trapezoid

Answer 8

A quadrilateral in which one pair of opposite sides are parallel but the other opposite sides are not parallel. If the two parallel sides are equal in length, the trapezoid is referred to as an isosceles trapezoid.

Area = ((b1+b2)*h)/2

Question 9

Sum of Interior Angles of a Polygon

Answer 9

Sum of Interior = (n-2)*180, n is number of sides

A “regular” polygon has all equal interior angles and all equal sides

Question 10

Hexagon

Answer 10

Reg Hexagon

• 6 equal sides and angles
• Interior angles sum to 720
• One interior angle is 120
• It can be divided into 6 equilateral triangles

Area of reg hex = 3sqrt(3)s^2/2, s is the length of any side
Approx. = 2.6s^2
Area = 1.5ds, d is distance between any two parallel sides, s is length of any side

Question 11

Circles

Answer 11

Circumference = 2pi*r or pi*d
Area = pi*r^2

Ratios:
(arc angle/360):(arc length/circumference):(area of arc/total area)

Inscribed angles:
The inscribed angle is equal to half the degree of measure of the arc that it intercepts. Or the arc angle is double the inscribed angle it intercepts.

Inscribed Triangles:
When a triangle is inscribed in a circle, if one side of the triangle is also the diameter of the circle, then the triangle is a right triangle, with the 90 vertex opposite the diameter. Or if a right triangle is inscribed in a circle, then its hypotenuse is the diameter of the circle.

When an equilateral triangle is inscribed in a circle, the triangle divides the circumference into three arcs.

When a regular polygon (all sides are equal) is inscribed in a circle, the polygon divides the circle in arcs of equal lengths.

When a square or rectangle is inscribed in a circle, the diagonal of the square/rectangle is also the diameter of the circle.

Question 12

Solid Rectangles & Cubes

Answer 12

The longest line segment of rectangular solid or cube is the diagonal (one corner to the opposite corner, through the middle)

Extended Pythagorean Theorem (longest line segment):
Rectangle = d^2=l^2+w^2+h^2
Cube = d = s*root(3)

Volume = l x w x h

Surface Area (6 faces total)
Cube = 6s^2
Rectangle = 2LW + 2LH + 2WH

Question 13

Cylinders

Answer 13

Volume = pi*r*^2*h
SA = 2*pi*r*h + 2*pi*r^2

Question 14

Sum of Exterior Angles

Answer 14

When taking one exterior angle at each vertex, the sum of the measures of the exterior angles will always equal 360.

An exterior angle is created by one side of a triangle and the extension of an adjacent side.

Question 15

Regular polygon inscribed in circle

Answer 15

A regular polygon inscribed in a circle will divide up the circumference in equal arc lengths.

Question 16

Tricky word problems

Answer 16

Watch out for “cube facing down” (means only 5 faces). Or bars “laid the same way” in a box (means only 1 orientation for the bars)

Question 17

Polynomials and Distances

Answer 17

Distances are always positive, use this to simplify polynomial expressions. Take the square root of (x+y)^2 = 2x^2 directly on both sides.