a. The pair of similar triangles are PSR and triangle PQS.
b. The triangles PSR and PQS are similar by the Angle-Angle (AA) postulate.
c. The value of PR = 8.06
Part A: The pair of similar triangles in the given figure is triangle PSR and triangle PQS.
Part B: The triangles PSR and PQS are similar by the Angle-Angle (AA) postulate. In both triangles, angle P is a 90-degree angle (right angle), and they share angle PSR and angle PQS. Therefore, the two triangles have two congruent angles, making them similar by the AA postulate.
Part C: To find the length of segment RP, we can use the Pythagorean Theorem. Given that RS = 4 and RQ = 16, we need to find the length of RP. Using the Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides, we have:
Pythagoras Theorem on AQPR
(16)²= (PQ)² + (PR)²
256 = PQ² + PR² - (i)
P.T of ARPS
(PR)²= (SP)² + (4) 2 2
(PR) = SP² + 16 - (ii)
P.T OF A SPR
(QP)² - (SP)+(SQ)²
(QP)² = SP² + SQ²
(QP)² = SP²+(12)²
(QP)² = SP² + 144 - (iii)
Adding eq(ii) and (iii)
(PR)² + (QP)² = SP² + SP² + 16 +144
(QR)²= 12SP²+160
(16)2 = 2(SP²+80)
128 = SP² +80
SP² = 128-80
SP2 = 48
√SP = √48
SP = 6.9 or 7
Now,
(PR) = (SP)²+(R3)
(PR) = (7)² + (4)
(PB)² = 49+16
PR)=V65
PR = 8.06