Unit 7 Math Test

45-45-90 Theorem

• In a 45-45-90 triangle, the hypotenuse is (square root of 2) times as long as a leg.

• “a” root 2

What is a 45-45-90 triangle?

An isosceles right triangle

How are the sides of a 30-60-90 triangle related?

• The hypotenuse is twice the length of the shorter leg.

• The longer leg is equal to the shorter leg times the square root of 3.

What is the tangent ratio?

A specific ratio between two side lengths

What is the equation for the tangent ratio?

• Tan (any angle) = opposite / adjacent

• Or TOA

What conclusions can be made about isosceles right triangles?

• The legs are congruent

• The angles are 45-45-90

• The hypotenuse = leg (root 2)

30-60-90 Theorem

In a 30-60-90 triangle, the hypotenuse is twice as long as the short leg and the longer leg is square root of 3 times as long as the short leg.

What are the side length equations in a 30-60-90 triangle?

• The hypotenuse is 2a

• The legs are a times (the square root of 3) and a

1²

1

2²

4

3²

9

4²

16

5²

25

11²

121

12²

144

13²

169

14²

196

15²

225

6²

36

7²

49

8²

64

9²

81

10²

100

16²

256

17²

289

18²

324

19²

361

20²

400

Pythagorean theorem

a² + b² = c²

True or False: The Pythagorean theorem only works with right triangles

True

Geometric mean

The value x that satisfies the proportion a/x = x/b where a and b are two positive numbers.

Similar right triangles inside right triangles

If we draw an altitude from the hypotenuse of a right triangle, 2 similar triangles are formed. They are similar to each other and the original triangles.

Geometric mean corollary/corollary 1

The length of the altitude is the geometric mean between the segments of the hypotenuse.

Geometric mean corollary equation

Left / Height = Height / Right

Corollaries as proportions/Corollary 2

Each leg of the geometric mean between the hypotenuse and the shadow (segment of the hypotenuse adjacent to that leg.

Corollaries as proportions equation

Whole / Leg = Leg / Shadow

Converse of Pythagorean theorem

If a² + b² = c², then triangle ABC is a right triangle.

If c² = a² + b², then the triangle is _____

right

If c² < a² + b², then the triangle is _____

acute

If c² > a² + b², then the triangle is _____

obtuse

Pythagorean triple

A set of 3 positive integers that satisfy the equation and always work.

In a 30-60-90 triangle, what value is across from the 30?

a

In a 30-60-90 triangle, what value is across from the 60?

a root 3

In a 30-60-90 triangle, what value is across from the 90?

2a

In a 45-45-90 triangle, what value is across from the 90?

a root 2

What are the values of the sides of a 45-45-90 triangle that are across from the 45s?

a