7.3 Use Similar Right Triangles
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1 7.3 Use Similar Right Triangles oal p Use properties of the altitude of a right triangle. Your Notes THEOREM 7. If the altitude is drawn to the hpotenuse of a right triangle, then the two triangles formed are to the original triangle and to each other. n n, n n, and n n. Example 1 Identif similar triangles Identif the similar triangles in the diagram. Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. H H H n, n, n heckpoint omplete the following exercise. 1. Identif the similar triangles in the diagram. N P 13 L 12 M 184 Lesson 7.3 eometr Notetaking uide opright Holt Mcougal. ll rights reserved.
2 7.3 Use Similar Right Triangles oal p Use properties of the altitude of a right triangle. Your Notes THEOREM 7. If the altitude is drawn to the hpotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. n, n, n, n, and n, n. Example 1 Identif similar triangles Identif the similar triangles in the diagram. Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. H H H n, n H, n H heckpoint omplete the following exercise. 1. Identif the similar triangles in the diagram. N P 13 L M 12 nnpl, nlpm, nnlm 184 Lesson 7.3 eometr Notetaking uide opright Holt Mcougal. ll rights reserved.
3 Example 2 ind the length of the altitude to the hpotenuse Stadium cross section of a group of seats at a stadium shows a drainage pipe that leads from the seats to the inside of the stadium. What is the length of the pipe? 20 ft 13 ft Step 1 Identif the similar triangles and sketch them. 20 ft 20 ft 13 ft Notice that if ou tried to write a proportion using n and n, there would be two unknowns, so ou would not be able to solve for x. n, n, n Step 2 ind the value of x. Use the fact that n, n to write a proportion. orresponding side lengths of similar triangles are in proportion. x Substitute. x x < The length of the pipe is about ross Products Propert pproximate. feet. heckpoint omplete the following exercise. 2. Identif the similar triangles. Then find the value of x. 6 x 8 E opright Holt Mcougal. ll rights reserved. Lesson 7.3 eometr Notetaking uide 18
4 Example 2 ind the length of the altitude to the hpotenuse Stadium cross section of a group of seats at a stadium shows a drainage pipe that leads from the seats to the inside of the stadium. What is the length of the pipe? 20 ft 13 ft Step 1 Identif the similar triangles and sketch them. 20 ft 20 ft 13 ft Notice that if ou tried to write a proportion using n and n, there would be two unknowns, so ou would not be able to solve for x. n, n, n Step 2 ind the value of x. Use the fact that n, n to write a proportion. x Ï 13 orresponding side lengths of similar triangles are in proportion. Substitute. ( Ï 13 ) x 20(30) ross Products Propert x < 16.6 pproximate. The length of the pipe is about 16.6 feet. heckpoint omplete the following exercise. 2. Identif the similar triangles. Then find the value of x. n, ne, ne; x x 8 E opright Holt Mcougal. ll rights reserved. Lesson 7.3 eometr Notetaking uide 18
5 Example 3 Use a geometric mean Notice that ne and ne both contain the side with length, so these are the similar pair of triangles to use to solve for. ind the value of. Write our answer in simplest radical form. Write a proportion. length of hp. of ne 4 E length of shorter leg of ne 1 Substitute. 2 ross Products Propert Ï Take positive square roots. Ï Simplif. THEOREM 7.6: EOMETRI MEN (LTITUE) THEOREM In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. The length of the altitude is the of the lengths of the two segments. THEOREM 7.7: EOMETRI MEN (LE) THEOREM In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. The length of each leg of the right and triangle is the geometric mean of the lengths of the hpotenuse and the segment of the hpotenuse that is to the leg. 186 Lesson 7.3 eometr Notetaking uide opright Holt Mcougal. ll rights reserved.
6 Notice that ne and ne both contain the side with length, so these are the similar pair of triangles to use to solve for. Example 3 Use a geometric mean ind the value of. Write our answer in simplest radical form. Write a proportion. length of hp. of ne length of hp. of ne length of shorter leg of ne length of shorter leg of ne E Substitute ross Products Propert Ï 60 Take positive square roots. 2 Ï 1 Simplif. THEOREM 7.6: EOMETRI MEN (LTITUE) THEOREM In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. THEOREM 7.7: EOMETRI MEN (LE) THEOREM In a right triangle, the altitude from the right angle to the hpotenuse divides the hpotenuse into two segments. The length of each leg of the right and triangle is the geometric mean of the lengths of the hpotenuse and the segment of the hpotenuse that is adjacent to the leg. 186 Lesson 7.3 eometr Notetaking uide opright Holt Mcougal. ll rights reserved.
7 Example 4 ind a height using indirect measurement Overpass To find the clearance under an overpass, ou need to find the height of a concrete support beam. 6.9 ft You use a cardboard square to line up the top and bottom of the beam. Your friend measures the vertical distance from the ground to our ee and the distance from ou to the beam. pproximate the height of the beam. ft Theorem 7.6, ou know that mean of and. is the geometric Write a proportion. x < Solve for x. So, the clearance under the overpass is 1 x < 1 feet. heckpoint omplete the following exercises. 3. ind the value of. Write our L answer in simplest radical form. K 24 M 9 4. The distance from the ground to Larr s ees is 4. feet. How far from the beam in Example 4 would he have to stand in order to measure its height? Homework opright Holt Mcougal. ll rights reserved. Lesson 7.3 eometr Notetaking uide 187
8 Example 4 ind a height using indirect measurement Overpass To find the clearance under an overpass, ou need to find the height of a concrete support beam. 6.9 ft You use a cardboard square to line up the top and bottom of the beam. Your friend measures the vertical distance from the ground to our ee and the distance from ou to the beam. pproximate the height of the beam. ft Theorem 7.6, ou know that 6.9 is the geometric mean of x and. x Write a proportion. x < 9. Solve for x. So, the clearance under the overpass is 1 x < feet. heckpoint omplete the following exercises. 3. ind the value of. Write our L answer in simplest radical form. 6 Ï 6 K 24 M 9 Homework 4. The distance from the ground to Larr s ees is 4. feet. How far from the beam in Example 4 would he have to stand in order to measure its height? about 6.7 feet opright Holt Mcougal. ll rights reserved. Lesson 7.3 eometr Notetaking uide 187
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