Similar Right Triangles

9.3 Similar Right Triangles Core Concepts Geometric Mean The geometric mean of two positive numbers a and b is the positive number x that   satisfies a/x = x/b So cross multiplied x = √ab Theorems Theorem 9.6 Right Triangle Similarity Theorem 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.   ΔCDB ~ ΔABC and ΔACD ~ ΔABC and ΔACB ~ ΔACD Theorem 9.7 Geometric Mean (Altitude) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse.   AD/CD = CD/BD Theorem 9.8 Geometric Mean (Leg) Theorem In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.   The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the   leg. a/b = b/c a/c = c/b 9.3 Extra Practice In Exercises 1 and 2, Identify the similar triangles. 1. ΔJHI ~ ΔJHK ΔJHI ~ ΔJKI ΔJHK ~ ΔJKI 2. ΔOPM ~ ΔPMN ΔPMN ~ ΔOMN ΔOMN ~ ΔOPM In Exercises 3 and 4, find the geometric mean of the two numbers. 3. 2 and 6 2/x = x/6 2*6 = x^2 12 = x^2 √12 = x 2√3 = x 4. 5 and 75 5/x = x/75 5*75 = x^2 375 = x^2 √375 = x 15√5 = x In Exercises 5-8, find the value of the variable. 5. use alt as geo mean 9/x = x/16 9*16 = x^2 144 = x^2 12 = x 6. use leg as geo mean 9/y = y/11 9*11 = y^2 99 = y^2 3√11 = y 7. use leg as geo mean 7/49 = 49/t 7*49 = 49t 343 = t 8. use alt as geo mean 3/x = x/6 3*6 = x^2 18 = x^2 3√2 = x