10-2 Special Right Triangles

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Lesson Presentation

10-2 Special Right Triangles

Do Now #13

For Exercises 1 and 2, find the value of x. Give your answer in simplest radical form.

1. 2. 2.

Simplify each expression.

3. 4.

10-2 Special Right Triangles

SWBAT

Justify and apply properties of 45°-45°-90° triangles.

Justify and apply properties of 30°- 60°- 90° triangles.

By the end of today’s lesson,

Connect to Mathematical Ideas (1)(F)

10-2 Special Right Triangles

A diagonal of a square divides it into two congruent isosceles right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle.

A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90°triangle.

10-2 Special Right Triangles

Explore:

150 yd

150 yd

300 yd

300 yd

300 yd

The distance from the dorm to the library is twice the given distance from the dorm to the dining hall, so each side of the quad is 300 yd. Using the Pythagorean Theorem, it is about 424.3 yd from the dorm to the science lab.

d

10-2 Special Right Triangles

10-2 Special Right Triangles

What is the value of each variable ?

Example 1: Finding the Length of the Hypotenuse

10-2 Special Right Triangles

What is the value of x ?

Example 2: Finding the Length of the Leg

45o– 45o– 90o Triangle Theorem

Substitute.

Divide both side by 2.

Multiply by a form of 1 to rationalize the denominator.

⇒ Simplify.

10-2 Special Right Triangles

Softball A high school softball diamond is a square. The distance from base to base is 60 ft. To the nearest foot, how far does a catcher throw the ball from home plate to second base?

Example 3: Finding Distance

The distance d is the length of the hypotenuse of a 45o – 45o – 90o triangle.

𝑑 = 60 2

𝑑 ≈ 84.85281374 Use a calculator.

The catcher throws the ball about 85 ft. from home plate to second base.

10-2 Special Right Triangles

10-2 Special Right Triangles

What is the value of each variable ?

Example 4: Use the Length of One Side

10-2 Special Right Triangles

An escalator lifts people to the second floor of a building, 25 ft. above the first floor. The escalator rises at a 30o

angle. To the nearest foot, how far does a person travel from the bottom to the top of the escalator ?

Example 5: Apply Mathematics (1)(A)

30o – 60o – 90o Triangle Theoremhypotenuse = 2 ⦁ shorter leg

𝑑 = 2 ⦁ 25

𝑑 = 50

∴ A person would travel 50 ft.from the bottom to the top of the escalator.

10-2 Special Right Triangles

Got It ? Solve With Your Partner

Problem 1 Finding the Length of the Hypotenuse.

What is the length of the hypotenuse of a

45o – 45o– 90o triangle with leg length 𝟓 𝟑 ?

5 6

10-2 Special Right Triangles

Got It ? Solve With Your Partner

Problem 2 Finding the Length of the leg.

a. The length of the hypotenuse of a 45o – 45o– 90o

triangle is 10. What is the length of one leg ?

5 2

b. In problem 2, why can you multiply ?

2

2= 1, so multiplying by

2

2is the same as

multiplying by 1.

10-2 Special Right Triangles

Got It ? Solve With Your Partner

Problem 3 Finding the Distance

You plan to build a path along one diagonal of a 100 ft-by-100ft square garden. To the nearest foot, how long will the path be?

141 feet

10-2 Special Right Triangles

Got It ? Solve With Your Partner

Problem 4 Using the Length of One Side

What is the value of f in simplest radical form ?

𝑓 =10 3

3

10-2 Special Right Triangles

Closure: Communicate Mathematical Ideas (1)(G)

What are special right triangle ?

They are 30o – 60o – 90o triangles and 45o – 45o – 90o triangles. They are studied because their special properties can be used as shortcuts for finding the lengths of the sides of these triangles.

10-2 Special Right Triangles

Exit Ticket: Check Your Understanding

Find the values of the variables. Give your answers in simplest radical form.

1. 2. 3.