1. Are these triangles similar? If so, give the reason. ANSWER Yes; the AA Similarity Postulate 2. Find x. ANSWER 50

Use special relationships in right triangles. Target Use special relationships in right triangles. You will… Use properties of the altitude of a right triangle.

Vocabulary Similar Triangles Formed by the Altitude to the Hypotenuse Theorem 7.5 – If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Triangles similar by AA~ CBD ~ ABC ACD ~ ABC CBD ~ ACD AB CB = BC BD CD AC AB AC = BC CD AD AC CB = CD BD AD

x a = b x2 = ab x = √ab Vocabulary Geometric Mean (between numbers a and b) – The positive value x such that means x a = b extremes x2 = ab x = √ab

Vocabulary Geometric Mean Altitude Theorem 7.6 – In a right triangle, the altitude from the right angle to the hypotenuse is the geometric mean between the sections of the hypotenuse it divides. CD BD = AD Geometric Mean Leg Theorem 7.7 – In a right triangle, either leg is the geometric mean between the section of the hypotenuse it is nearest and the whole hypotenuse. CB AB = DB AC AB = AD

EXAMPLE 1 Identify similar triangles Identify the similar triangles in the diagram. SOLUTION Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. TSU ~ RTU ~ RST

EXAMPLE 2 Find the length of the altitude to the hypotenuse Swimming Pool The diagram below shows a cross-section of a swimming pool. What is the maximum depth of the pool?

EXAMPLE 2 Find the length of the altitude to the hypotenuse SOLUTION STEP 1 Identify the similar triangles and sketch them. RST ~ RTM ~ TSM

Find the length of the altitude to the hypotenuse EXAMPLE 2 Find the length of the altitude to the hypotenuse STEP 2 Find the value of h. Use the fact that RST ~ RTM to write a proportion. ST TM = SR TR Corresponding side lengths of similar triangles are in proportion. 64 h = 165 152 Substitute. 165h = 64(152) Cross Products Property h 59 Solve for h. STEP 3 Read the diagram. You can see that the maximum depth of the pool is h + 48, which is about 59 + 48 = 107 inches. The maximum depth of the pool is about 107 inches.

GUIDED PRACTICE for Examples 1 and 2 Identify the similar triangles. Then find the value of x. 1. EGF ~ GHF ~ EHG ; 12 5 ANSWER 2. LMJ ~ MKJ ~ LKM ; 60 13 ANSWER

EXAMPLE 3 Use a geometric mean Find the value of y. Write your answer in simplest radical form. SOLUTION STEP 1 Draw the three similar triangles.

length of shorter leg of RQS length of shorter leg of RPQ EXAMPLE 3 Use a geometric mean STEP 2 Write a proportion. length of hyp. of RQS length of hyp. of RPQ = length of shorter leg of RQS length of shorter leg of RPQ y 9 = 3 Substitute. 27 = y2 Cross Products Property Take the positive square root of each side. 27 = y 3 3 = y Simplify.

EXAMPLE 4 Find a height using indirect measurement Rock Climbing Wall To find the cost of installing a rock wall in your school gymnasium, you need to find the height of the gym wall. You use a cardboard square to line up the top and bottom of the gym wall. Your friend measures the vertical distance from the ground to your eye and the distance from you to the gym wall. Approximate the height of the gym wall.

Find a height using indirect measurement EXAMPLE 4 Find a height using indirect measurement SOLUTION By Theorem 7.6, you know that 8.5 is the geometric mean of w and 5. 8.5 w = 5 Write a proportion. w 14.5 Solve for w. So, the height of the wall is 5 + w 5 + 14.5 = 19.5 feet.

GUIDED PRACTICE for Examples 3 and 4 3. In Example 3, which theorem did you use to solve for y ? Explain. Theorem 7.7; This was used to set the ratios of the hypotenuse of the large triangle to the shorter leg and the hypotenuse of the small triangle to the shorter leg equal to each other. ANSWER

GUIDED PRACTICE for Examples 3 and 4 4. Mary is 5.5 feet tall. How far from the wall in Example 4 would she have to stand in order to measure its height? ANSWER about 8.93 ft