In the given triangle abc, angle a is 90° and segment ad is perpendicular to segment bc.

the figure shows triangle abc with right angle at a and segment ad. point d is on side bc.

part a: identify a pair of similar triangles. (2 points)

part b: explain how you know the triangles from part a are similar. (4 points)

part c: if db = 9 and dc = 4, find the length of segment da. show your work. (4 points)

answer:

  part a: δacd ~ δbad

  part b: aa similarity

  part c: da = 6

step-by-step explanation:

it can to draw a diagram. (see attached.)

part a and b:

any pair of triangles in the figure are similar triangles. any of them is a right triangle, and each will have an acute angle of the same measure as an acute angle in the other.

for example, δacd ~ δbca because both are right triangles and both share angle c.   (aa similarity)

likewise, δbad ~ δbca because both are right triangles and both share angle b. (aa similarity)

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it takes another step to show δacd ~ δbad. you can either do it using the transitive property of similarity with the two similarity statements above, or you can show the triangles contain acute angles of equal measure.

you know ∠dac and ∠dca are complementary because they are the acute angles in right triangle acd. you know that ∠dac and ∠dab are complementary, because their sum is ∠cab, a right angle. by the transitive property of congruence, then, ∠dca ≅ ∠dab. hence, δacd ~ δbad by aa similarity.

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part c:

δacd ~ δbad, so

  dc/da = da/db . . . ratios are proportional in similar triangles

  dc·db = da² . . . . cross multiply

  4·9 = 36 = da² . . . substitute given values

  da = √36 = 6 . . . . solve for da

the length of segment da is 6.