PLEASE HELP Part C: If RS = 4 and RQ = 16, find the length of segment RP. Show your work. (4 points)

A. \triangle RSP \sim \triangle RPQ

B. All angles are same, so the triangles are similar.

C. RP = 8

Step-by-step explanation:

Given right angled \triangle PQR in which

\angle P =90^\circ

Line segment PS is perpendicular to QR and S is on the line QR.

Kindly refer to the attached diagram for the given dimensions and angles.

Part A. \triangle RSP \sim \triangle RPQ

Part B. Let us consider the two triangles: \triangle RSP \ and\ \triangle RPQ.

\angle RPQ = 90^\circ = \angle RSP

and \angle R is common to both the triangles.

Two angles of two triangles are equal to other other.

So, third angle will also be equal to each other.

\therefore by AAA similarity:

\triangle RSP \sim \triangle RPQ

Part C:

The ratio of corresponding sides of two triangles is equal.

So,

\dfrac{RP}{RQ} = \dfrac{RS}{RP}\\\Rightarrow RP^2 = 4 \times 16\\\Rightarrow RP = \sqrt{64}\\\Rightarrow \bold{RP = 8}

Seth is using the figure shown below to prove the Pythagorean Theorem using triangle similarity: In

Thanks