Question

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.

The figure shows triangle PQR with right angle at P and segment PS. Point S is on side 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)

Answer 1

posting on behalf of @iSuckAtMath101

Answer 2

Alright, so we know triangle QPR is a right triangle. can you find another similar triangle which is a right triangle?

Answer 3

psq?

Answer 4

wait no

Answer 5

QPR?

Answer 6

correct, it would be QSP

So you already have the answer to part A.

△QPR ~ △QSP

Answer 7

understood?

Answer 8

yes

Answer 9

ok, so now lets prove how these two triangles are similar. are you familar with the AA similarity theorem?

Answer 10

angle angle?

Answer 11

yes, can you define it?

Answer 12

when two angles are congruent it means that the triangles are similar?

Answer 13

correct,

The formal definition is: `In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.`

So now, what two angles in these two triangles are similar? I suggest you to draw it out so you can have a visual of how the triangles look like.

Answer 14

q and r?

Answer 15

Are you sure...? look at this drawing, which angles are similar?

Answer 16

p and s?

Answer 17

thats right, can you find another pair?

Answer 18

r and p

Answer 19

or q and q

Answer 20

yep, those can work, so for part B, just state 2 pairs of angles that are similar and your reasoning will be based off of the AA similarity theorem

Answer 21

alright

Answer 22

Part B: angles P and S, and R and P are congruent because of the AA similarity theorem

Answer 23

Now for part C, we will have to set up a proportion with the side lengths that are given...

do you have an idea of what the proportion will be/look like, or do you want me to walk you through it?

Answer 24

you can walk me through it please

Answer 25

alright,

so we are given RS (which is 4) and RQ (which is 16) and we need to find RP

Our proportion will look like:

\[\frac{ RQ }{ RP } = \frac{ RP }{ RS }\]

it might seem hard but just trust me here... just substitute the variables that are given and solve...

\[\frac{ 16 }{ RP } = \frac{ RP }{ 4 }\]

its easier to look at it like this

\[\frac{ 16 }{ x } = \frac{ x }{ 4 }\]

Now we just solve for x, can you do that? `(if you are having difficulty, just remember to cross multiply)`

Answer 26

64 = x(squared)?
then square root of 64 = 8?

Answer 27

yep, you got it, great job :D

`x = 8 which means RP = 8`

just remember to show your work and you'll be done :)

Answer 28

ty so much, and ty for not giving me the answer and making me work for it lol. It helped me and now I know how to do it!!