Geometry Chapter 7

7-4: SPECIAL RIGHT TRIANGLES

Warm-Up

Simplify the following.

1.) 10 × 30 2.) 45

5

3.) 88

84.) 3 × 27

Special Right Triangles

Objective: Students will be able to use the relationships amongst the

sides in special right triangles to find side lengths.

Agenda

45° − 45° − 90° Triangles

30° − 60° − 90° Triangles

Examples

45° − 45° − 90° Triangles

Definition

A 45° − 45° − 90° Triangle is an isosceles

right Triangle, with 45° as the measures

of both the other two angles.

45°

45°

Hypotenuse

Leg

Leg

45° − 45° − 90° Triangles

Definition

A 45° − 45° − 90° Triangle is an isosceles

right Triangle, with 45° as the measures

of both the other two angles.

Knowledge Connection

Both Legs in this triangle are congruent.

45°

45°

Hypotenuse

Leg

Leg

45° − 45° − 90° Theorem

Theorem 7.8: In a 45° − 45° − 90° right triangle, the hypotenuse is 2times as long as a leg.

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

45°

45°

𝒄𝒂

𝒃

Hypotenuse

Leg

Leg

45° − 45° − 90° Examples

Find the value of x.

45° − 45° − 90° Examples

Find the value of x.

Hypotenuse

Leg

45°

45° − 45° − 90° Examples

Find the value of x.

Hypotenuse

LegSolution:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

𝑥 = 12 × 2

𝒙 = 𝟏𝟐 𝟐

45°

45° − 45° − 90° Examples

Find the value of x.

45° − 45° − 90° Examples

Find the value of x.

Hypotenuse

Leg

Solution:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

8 = x × 2

45°

45° − 45° − 90° Examples

Find the value of x.

Hypotenuse

Leg

Solution:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

8 = x × 2

𝑥 =8

2×

2

2=8 2

2

𝒙 = 𝟒 𝟐

45°

45° − 45° − 90° Examples

Find the values of x and y.

45° − 45° − 90° Examples

Find the value of x and y.

Hypotenuse

Leg

45°

Leg

45° − 45° − 90° Examples

Find the value of x and y.

Hypotenuse

Leg

For x:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

2 6 = x × 2

𝑥 =2 6

2

𝒙 = 𝟐 𝟑

45°

Leg

45° − 45° − 90° Examples

Find the value of x and y.

Hypotenuse

Leg

For x:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

2 6 = x × 2

𝑥 =2 6

2

𝒙 = 𝟐 𝟑

45°

Leg

For y:

In a 45° − 45° − 90°triangle, the Legs have

the same length.

Therefore, 𝒚 = 𝟐 𝟑

45° − 45° − 90° Examples

Find the value of x.

𝟖

𝟖

𝒙

45° − 45° − 90° Examples

Find the value of x.

𝟖

𝟖

𝒙

Hypotenuse

Leg

Leg

45° − 45° − 90° Examples

Find the value of x.

For x:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

𝑥 = 8 × 2

𝒙 = 𝟖 𝟐𝟖

𝟖

𝒙

Hypotenuse

Leg

Leg

30° − 60° − 90° Triangles

Definition

A 30° − 60° − 90° is a right triangle with 30°and 60° as its other angle measures.

Shorter Leg

Longer Leg

Hypotenuse

30° − 60° − 90° Triangles

Definition

A 30° − 60° − 90° is a right triangle with 30°and 60° as its other angle measures.

Knowledge Connection

The leg Opposite the 30° angle is called

the Shorter Leg.

The Leg Opposite the 60° angle is called

the Longer Leg. Shorter Leg

Longer Leg

Hypotenuse

30° − 60° − 90° Theorem

Theorem 7-9: In a 30° − 60° − 90° right triangle, the hypotenuse is

twice as long as the shorter leg, and the longer leg is 3 times as

long as a shorter leg.

𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

𝐿. 𝐿. = 𝑆. 𝐿. × 3

𝒄

𝒂

𝒃

Shorter Leg

Longer Leg

Hypotenuse

30° − 60° − 90° Examples

Find the values of x and y.

30° − 60° − 90° Examples

Find the values of x and y.

Shorter Leg

Hypotenuse

Longer Leg

30° − 60° − 90° Examples

Find the values of x and y.

Shorter Leg

Hypotenuse

Longer Leg

For x:𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

𝑥 = 6 × 2

𝒙 = 𝟏𝟐

30° − 60° − 90° Examples

Find the values of x and y.

Shorter Leg

Hypotenuse

Longer Leg

For x:𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

𝑥 = 6 × 2

𝒙 = 𝟏𝟐

For y:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

𝑦 = 6 × 3

𝒚 = 𝟔 𝟑

30° − 60° − 90° Examples

Find the values of x and y.

𝒙

𝒚

𝟐𝟎

𝟔𝟎°

30° − 60° − 90° Examples

Find the values of x and y.

𝒙

𝒚

𝟐𝟎

𝟔𝟎°Shorter Leg

Hypotenuse

Longer Leg

30° − 60° − 90° Examples

Find the values of x and y.

𝒙

𝒚

𝟐𝟎

𝟔𝟎°Shorter Leg

Hypotenuse

Longer Leg For x:

𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

20 = 2x

𝒙 = 𝟏𝟎

30° − 60° − 90° Examples

Find the values of x and y.

𝒙

𝒚

𝟐𝟎

𝟔𝟎°Shorter Leg

Hypotenuse

Longer Leg For x:

𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

20 = 2x

𝒙 = 𝟏𝟎

For y:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

𝑦 = 10 × 3

𝒚 = 𝟏𝟎 𝟑

30° − 60° − 90° Examples

Find the values of x and y.

30° − 60° − 90° Examples

Find the values of x and y.

Shorter Leg

Hypotenuse

Longer Leg

30° − 60° − 90° Examples

Find the values of x and y.

Shorter Leg

Hypotenuse

Longer Leg

For x:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

8 = x 3

𝑥 =8

3

𝑥 =8

3×

3

3=𝟖 𝟑

𝟑

30° − 60° − 90° Examples

Find the values of x and y.

Shorter Leg

Hypotenuse

Longer Leg

For x:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

8 = x 3

𝑥 =8

3

𝑥 =8

3∗

3

3=𝟖 𝟑

𝟑

For y:𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

𝑦 = 𝑥 × 2

𝑦 = 2 ×8 3

3

𝒚 =𝟏𝟔 𝟑

𝟑

30° − 60° − 90° Examples

Find the values of x and y.

𝟔𝟔

𝒙

𝟑𝟑

30° − 60° − 90° Examples

Find the values of x and y.

𝟔𝟔

𝒙

𝟑𝟑Shorter Leg

Hypotenuse

Longer Leg

30° − 60° − 90° Examples

Find the values of x and y.

𝟔𝟔

𝒙

𝟑𝟑

For x:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

x = 3 × 3

𝒙 = 𝟑 𝟑

Shorter Leg

Hypotenuse

Longer Leg

Final Practice: Both Triangles

Find the values of the variables in the given diagram.

For u:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

8 2 = u × 2

𝑢 =8 2

2

𝒖 = 𝟖

For v:

In a 45° − 45° − 90°triangle, the Legs have

the same length.

Therefore, 𝐯 = 𝟖

Final Practice: Both Triangles

Find the values of the variables in the given diagram.

𝒏

𝒎

𝟏𝟎𝟒𝟓°

For m:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

10 = m× 2

𝑚 =10

2

𝒎 = 𝟓

For n:

In a 45° − 45° − 90°triangle, the Legs have

the same length.

Therefore, 𝐧 = 𝟓

Final Practice: Both Triangles

Find the values of the variables in the given diagram.

For a:

𝐻𝑦𝑝 = 𝐿𝑒𝑔 × 2

𝑎 = 2 2 × 2

𝑎 = 2(2)

𝒂 = 𝟒

For b:

In a 45° − 45° − 90°triangle, the Legs have

the same length.

Therefore, 𝐛 = 𝟐 𝟐

Final Practice: Both Triangles

Find the values of the variables in the given diagram.

For u:

𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

u = 2 × 2

𝒖 = 𝟒

For v:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

𝑦 = 2 × 3

𝒚 = 𝟐 𝟑

Final Practice: Both Triangles

Find the values of the variables in the given diagram.

For y:

𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

8 5 = 2y

𝒚 = 𝟒 𝟓

For y:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

𝑦 = 4 5 × 3

𝒚 = 𝟒 𝟏𝟓

Final Practice: Both Triangles

Find the values of the variables in the given diagram.

For a:

𝐻𝑦𝑝 = 𝑆. 𝐿. × 2

a = 11 × 2

𝒂 = 𝟐𝟐

For b:

𝐿. 𝐿. = 𝑆. 𝐿. × 3

11 3 = 𝑏 × 3

𝑏 =11 3

3

𝒃 = 𝟏𝟏