Geometry 7.6 Proportional Lengths

Proportional Lengths AC and XZ are divided proportionally if… X ABC YZ = BC XYAB YZ Example: = 2 9 4

Triangle Proportionality Thm. If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. big A small A whole A big B small B whole B = side C1 side C2 = = ==== whole A small A whole B small B side C1 side C2 big A small A big B small B big A whole A big B whole B whole A big Abig B small A small B big A big B whole A whole B All of these proportions, and their inverses, work. The key is to use the easiest one to solve each problem. Think of it as two separate similar triangles.

Corollary If three // lines intersect two transversals,… then they divide the transversals proportionally. a b c d = a b c d

Triangle Angle Bisector Thm. If a ray bisects an angle of a triangle,… then it divides the opposite side into segments proportional to the other two sides. a b c d = a b c d

Directions: Use the given information and the triangle above to find the missing segment. 3. GE = 15 DE = 27 HF = 20 DH = ? 4.DE = 20 HF = 4 DH = 12DG = ? E F G H D

Directions: Use the given information and the diagram to find x. 2. AB = 27BC = 18 DE = x + 10EF = x 3. AB = 25 – x BC = x DE = 16EF = 4 A B C D E F

Directions: is and angle bisector of. Find x. 2. RP = x PQ = 12.5RS = 8SQ = RP = 10 PQ = 20RS = xRQ = 15 P Q R S

HW P. 271 (1-7) P (1-14, 20, 21) Quiz Tomorrow

A few from the HW P. 272 #5, #14

An Example = = = == ==

Solve for x. (figure not to scale) x = x = x 5 4 Reduce by 3 times 4 equals x = 16

Solve for x x x + 10 = x x + 10 Reduce by 9. = 2 3 x x (x + 10) = 3x 2x + 20 = 3x x =

Solve for x x = x 18 - x Reduce by 3. = 4 5 x 18 - x 4(18 – x) = 5x 72 – 4x = 5x 9x = 72 x = 8 810