

3 and 27Ĭopyright © by Holt, Rinehart and Winston. If necessary, give the answer in simplest radical form. 1.įind the geometric mean of each pair of numbers. Write a similarity statement comparing the three triangles in each diagram. Similarity statement: 䉭ABC 䉭ADB 䉭BDC The geometric mean of two positive numbers is the positive square root of their product. The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. The cliff is about 142.5 + 5.5, or 148 ft high.Similarity in Right Triangles Altitudes and Similar Triangles Let x be the height of cliff above eye level. What is the height of the cliff to the nearest foot? (28)2 = 5.5x 28 is the geometric mean of 5.5 and y. The tree is about 38 + 1.6 = 39.6, or 40 m tall.Įxample 8-on your own! A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. h2 = (x)(y) 7.8 is the geometric mean of 1.6 and y. What is the height of the tree to the nearest meter? Let x be the height of the tree above eye level. Her eyes are about 1.6 m above the ground, and she is standing 7.8 m from the tree. Find the positive square root.Įxample 7 To estimate the height of a Douglas fir, Jan positions herself so that her lines of sight to the top and bottom of the tree form a 90º angle. a2 = (x)(c) v2 = (27 + 3)(3) v is the geometric mean of u + 3 and 3.

h2 = (x)(y) 92 = (3)(u) 9 is the geometric mean of u and 3. REMEMBER!!!Įxample 6: On your own! Find u, v, and w. Helpful Hint Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers. y2 = (4)(13) = 52 Find the positive square root. a2 = (x)(c) y is the geometric mean of 4 and 13. h2 = (x)(y) 62 = (x)(9) 6 is the geometric mean of 9 and x. All the relationships in red involve geometric means.Įxample 5: Finding side lengths in right triangles Find x, y, and z. You can use Theorem 8-1-1 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. of geometric mean Find the positive square root. of geometric mean x = 10 Find the positive square root.Įxample 4: On your own! Find the geometric mean of each pair of numbers.

So the geometric mean of a and b is the positive number x such that, or x2 = ab.Įxample 3: Finding geometric means Find the geometric mean of each pair of numbers. The geometric meanof two positive numbers is the positive square root of their product. In this case, the means of the proportion are the same number, and that number is the geometric mean of the extremes. By Theorem 8-1-1, ∆LJK ~ ∆JMK ~ ∆LMJ.ĭefinition 1 Consider the proportion. Sketch the three right triangles with the angles of the triangles in corresponding positions. By Theorem 8-1-1, ∆UVW ~ ∆UWZ ~ ∆WVZ.Įxample 2-On your own Write a similarity statement comparing the three triangles. W Z Example 1- Identifying Similar Right Triangle Write a similarity statement comparing the three triangles. Theorem 1 In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles. Write a similarity statement comparing the two triangles. 8-1 Similarity in Right Triangles 2/19/13īell Work 1.
