Special Right Triangles

45 – 45 – 90 Triangles

Special Right Triangles Directions

As you view this presentation, take notes and work out the practice problems.

When you get to the practice problem screens, complete the step in your notebook before continuing to the next slide.

45- 45- 90 Triangles• A 45 – 45 – 90 triangle is also

known as an isosceles right triangle.

• An isosceles right triangle is a right triangle with 2 equal sides or legs. (a = b)

• The 2 angles across from the equal sides each measure 45o. (angle A = angle B = 45o)

c

a

b

45o

45o

A

C B

45- 45- 90 Triangles

• Because the lengths of the 2 legs in 45 – 45 – 90 triangle are equal, the legs are usually labeled x.

• The hypotenuse in a 45-45-90 triangle is often labeled h.

hx

x

45o

45o

45- 45- 90 TrianglesFinding the Length of the Hypotenuse

• The Pythagorean Theorem can be used to find the length of the hypotenuse when given the length of the legs.

• x2 + x2 = h2

• 2x2 = h2

• 𝑥2 ∗ 2 = h2

• x 2 = h

• You can save a lot of time and work if you remember

h = x 2

hx

x

45o

45o

45- 45- 90 TrianglesPractice Problem 1

• Find the length of x

h

x = ?

5

45o

45o

45- 45- 90 TrianglesPractice Problem 1

• Find the length of x

• The two legs of a 45 – 45 – 90 triangle are equal so

x = 5 h

x = 5

5

45o

45o

45- 45- 90 TrianglesPractice Problem 1

• Find the length of h

h = ?

x= 5

5

45o

45o

45- 45- 90 TrianglesPractice Problem 1

• Find the length of h

• You can always use the Pythagorean Theorem to find the length of h.

• But if you remember the shortcut

h = x 2

x = 5

5

45o

45o

45- 45- 90 TrianglesPractice Problem 1

• Find the length of h

• You can always use the Pythagorean Theorem to find the length of h.

• But if you remember the shortcut

• h = 5 2

h = 5 2

x = 5

5

45o

45o

45- 45- 90 TrianglesFinding the Lengths of the Legs

• The Pythagorean Theorem can be used to find the lengths of the legs when given the length of the hypotenuse.

• x2 + x2 = h2

• 2x2 = h2

• x2 = ℎ2

2

• 𝑥2 = ℎ2

2

• x = ℎ

2*

2

2= ℎ 2

2

• (Remember to always rationalize the denominator)

hx

x

45o

45o

45- 45- 90 TrianglesFinding the Lengths of the Legs

You can save a lot of time and work if you remember

x = ℎ 2

2 hx =

ℎ 2

2

x = ℎ 2

2

45o

45o

45- 45- 90 TrianglesPractice Problem 2

• Find the length of x

h = 3

x = ?

x = ?

45o

45o

45- 45- 90 TrianglesPractice Problem 2

• Find the length of x

• You can always use the Pythagorean Theorem to find the lengths of the legs.

h = 3

x = ?

x = ?

45o

45o

45- 45- 90 TrianglesPractice Problem 2

• Find the length of x

• But if you remember the shortcut

• x = ℎ 2

2

h = 3

x = ℎ 2

2

45o

45o

x = ℎ 2

2

45- 45- 90 TrianglesPractice Problem 2

• Find the length of x

• But if you remember the shortcut

• x = ℎ 2

2

• Then x = 3 2

2

h = 3

x = 3 2

2

45o

45o

x = 3 2

2

45- 45- 90 TrianglesPractice Problem 3

• Find the length of x

h = 1

x = ?

x = ?

45o

45o

45- 45- 90 TrianglesPractice Problem 3

• Find the length of x

• You can always use the Pythagorean Theorem to find the lengths of the legs.

h = 1

x = ?

x = ?

45o

45o

45- 45- 90 TrianglesPractice Problem 3

• Find the length of x

• But if you remember the shortcut

• x = ℎ 2

2

h = 1

x = ℎ 2

2

45o

45o

x = ℎ 2

2

45- 45- 90 TrianglesPractice Problem 3

• Find the length of x

• But if you remember the shortcut

• x = ℎ 2

2

• Then x = 1 2

2=

2

2

h = 1

x = 2

2

45o

45o

x = 2

2

45- 45- 90 Trianglesin the Unit Circle

• In the Unit Circle:

• h = 1

• So remembering this shortcut for a 45 – 45 - 90 triangle will save you time and work.

• x = 2

2

h = 1

x = 2

2

45o

45o

x = 2

2