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PLEASE HELP!!! WILL MARK BRAINLIEST!!! Seth is using the figure shown below to prove the Pythagorean Theorem using triangle similarity:
In the given triangle PQR, angle P is 90° and segment PS is perpendicular to segment QR.
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 RS = 4 and RQ = 16, find the length of segment RP. Show your work. (4 points)

2 Answers

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The altitude on hypotenuse theorem states that the right angle formed by PS divides the original triangle into triangles that are similar to each other and also to the original larger triangle that contains both.

Draw this and RS is to PS as PS is to 12 or SQ

that makes PS^2=48 and PS= 4 sqrt(3).

RQ is to PQ as PQ is to 12 or SQ

this is 16 is to PQ as PQ is to 12

PQ^2=192 and PQ is sqrt (192) or 8 sqrt (3)

The line PS is 4 sqrt(3) and these are all 30-60-90 right triangles.

that is one way to get the answer for RP, which is half the hypotenuse of the large triangle, or 8,


the other way is that 16/RP =RP/4

RP^2=64, and RP=8.

Step-by-step explanation:
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The larger triangle ABC is similar to the smaller triangles as follows.

Triangle ABC ~ triangle DBA by AA similarity because they share angle B and angles A and D are right angles.

Triangle ABC ~ triangle DAC by AA similarity because they share angle C and angles A and D are right angles.

So you can set up the following proportions, seeing that the answers are involving either AC^2 or AB^2.

AB/BD = BC/AB When you cross multiply, you get AB^2 = BC times BD, which is the first answer listed.

Using the other proportion, AC/CD = BC/AC, when you cross multiply, you get AC^2 = BC times CD, which is not one of the answers listed.
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