Similar Right Triangles Lesson 13.3 Similar Right Triangles pp. 548-553

Objectives: 1. To prove that the altitude to the hypotenuse of a right triangle divides it into two right triangles, each similar to the original. 2. To define and apply geometric means. 3. To compute lengths of sides and related segments for right triangles by using proportions.

Theorem 13.4 An altitude drawn from the right angle to the hypotenuse of a right triangle separates the original triangle into two similar triangles, each of which is similar to the original triangle.

A C D B BDC ~ ABC ADB ~ BDC ADB ~ ABC

In the proportion , notice that the denominator of one ratio is the same as the numerator of the other ratio. When this happens, x is called the geometric mean. b x a =

For example, 8 is the geometric mean between 16 and 4 because = .

EXAMPLE 1 Find the geometric mean between 3 and 27. = x2 = 81 x =  81 x = 9

EXAMPLE 2 Find the geometric mean between 6 and 9. = x2 = 54 x =  54 x = 3 6

Practice: Find the geometric mean between 5 and 25. x 5 = x2 = 125 125 x ± = ≈ 11.2 5 x =

Practice: Find the geometric mean between 12 and 20.

Theorem 13.5 In a right triangle, the altitude to the hypotenuse cuts the hypotenuse into two segments. The length of the altitude is the geometric mean between the lengths of the two segments of the hypotenuse.

D C B A a b x BC DB AB or , b x a =

Theorem 13.6 In a right triangle, the altitude to the hypotenuse divides the hypotenuse into two segments such that the length of a leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to the leg.

D C B A AC DC BC = AC AD AB =

EXAMPLE 3 Given the measurements in HIJ, find x, y, and z. 4 16 16 x 4 = x2 = 64 x = 8

EXAMPLE 3 Given the measurements in HIJ, find x, y, and z. 4 16 20 y 4 = y2 = 80 y = 4 5

EXAMPLE 3 Given the measurements in HIJ, find x, y, and z. 4 16 20 z 16 = z2 = 320 z = 8 5

Practice: Given: Right JKL. with altitude to the Practice: Given: Right JKL with altitude to the hypotenuse, MK; LJ = 20, and MJ = 4, find KM. K J L M

Practice: Given: Right JKL. with altitude to the Practice: Given: Right JKL with altitude to the hypotenuse, MK; MJ = 4, and KJ = 6, find LJ. K J L M

Homework pp. 552-553

►A. Exercises Solve each proportion; assume that x is positive. 3. x 4 9 =

►A. Exercises Solve each proportion; assume that x is positive. 5. x 2 x – 3 =

7. AD = 15 units; DB = 5 units; find AC ►B. Exercises Given that ∆ABC is a right triangle and DC is an altitude to the hypotenuse, AB, find the length of the indicated sides. 7. AD = 15 units; DB = 5 units; find AC A D B C

9. AB = 32 units; DB = 6 units; find CD ►B. Exercises Given that ∆ABC is a right triangle and DC is an altitude to the hypotenuse, AB, find the length of the indicated sides. 9. AB = 32 units; DB = 6 units; find CD A D B C

11. AD = 6 units; AB = 10 units; find CD ►B. Exercises Given that ∆ABC is a right triangle and DC is an altitude to the hypotenuse, AB, find the length of the indicated sides. 11. AD = 6 units; AB = 10 units; find CD A D B C

13. AD = 11 units; DB = 5 units; find AC ►B. Exercises Given that ∆ABC is a right triangle and DC is an altitude to the hypotenuse, AB, find the length of the indicated sides. 13. AD = 11 units; DB = 5 units; find AC A D B C

15. AD = 12 units; AB = 18 units; find CB ►B. Exercises Given that ∆ABC is a right triangle and DC is an altitude to the hypotenuse, AB, find the length of the indicated sides. 15. AD = 12 units; AB = 18 units; find CB A D B C

■ Cumulative Review Which pairs of figures are similar? For each pair of similar figures, give the scale factor, k. 23. Two circles with radii 3 and 6

■ Cumulative Review Which pairs of figures are similar? For each pair of similar figures, give the scale factor, k. 24. Two rectangles: 6 by 9 and 8 by 12.

■ Cumulative Review Which pairs of figures are similar? For each pair of similar figures, give the scale factor, k. 25. Two rectangles: 6 by 8 and 16 by 18.

■ Cumulative Review Which pairs of figures are similar? For each pair of similar figures, give the scale factor, k. 26. Two regular tetrahedra with sides of length 9 and 6 respectively

■ Cumulative Review Which pairs of figures are similar? For each pair of similar figures, give the scale factor, k. 27. Two squares with sides of length s and t respectively.

Analytic Geometry Slopes of Perpendicular Lines

Theorem If two distinct nonvertical lines are perpendicular, then their slopes are negative reciprocals.

Z(x2, y1) X(x1, 0) Y(x3, 0) W(x2, 0) l1 l2

►Exercises Give the equation of the line perpendicular to the line described and satisfying the given conditions. 1. y = -4/3x + 5 with y-intercept (0, -8)

►Exercises Give the equation of the line perpendicular to the line described and satisfying the given conditions. 2. y = 2x – 1 and passing through (1, 4)

►Exercises Give the equation of the line perpendicular to the line described and satisfying the given conditions. 3. the line containing (2, 5) and (3, 4) at the first point.

►Exercises Give the equation of the line perpendicular to the line described and satisfying the given conditions. 4. y = 1/2x + 5, if their point of intersection occurs when x = 2