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p.1 Similar Right Triangles p.2 Similar Right Triangles Example: Finding the height of a roof. Investigation 1. Cut an index card along one of its diagonals. Label the vertices A, B, C. 2. On one of the right triangles, draw an altitude from the right angle to the hypotenuse. Cut along the altitude to form two right triangles. Roof Height: A roof has a cross section that is a right angle. The diagram shows the approximate dimensions of this cross section. 3. You should now have 3 right triangles. Compare the triangles. What special property do they share? Explain. A. Identify the similar triangles. (It may help to redraw them as separate triangles.) B. Find the height h of the roof using what you have learned about similar triangles. 4. What did you discover? Our next theorem states: 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. C ∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD A D B p.3 Similar Right Triangles p.4 Similar Right Triangles C A Write a proportion comparing sides CB, AB, and BD in the first and third triangles. D B In right ∆ABC, altitude CD is drawn to the hypotenuse, forming two smaller right triangles that are similar to ∆ABC. The triangles have been redrawn below as 3 separate pictures. Using colored pencils, shade the sides that are the same the same color. __________ is the geometric mean between _______ and ________. B C C B D A Write a proportion comparing sides AC, AB, and AD in the second and third triangles. D A C Write a proportion comparing sides AD, CD, and BD in the first two triangles. __________ is the geometric mean of ________ and ________. __________ is the geometric mean between _______ and ________. What do you think the geometric mean is in your own words? p.5 Similar Right Triangles p.6 Similar Right Triangles Geometric Mean Theorems Example 2: 2 C y 5 Solve for y. A D B In a right triangle, the _______________ from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the _______________ _______________ of the lengths of the two segments. BD CD CD AD In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each __________ of the right triangle is the _______________ _________ of the lengths of the hypotenuse and the segment of the hypotenuse that is _______________ to the leg. Application: MONORAIL TRACK To estimate the height of a monorail track, your friend holds a cardboard square at eye level. Your friend lines up the top edge of the square with the track and the bottom edge with the ground. You measure the distance from the ground to your friend’s eye and the distance from your friend to the track. AB AC AC AD AB CB CB DB Example 1: x 6 3 Solve for x. Hint: You must find XY in order to write a proportion using what we have learned.