Geometry : )

the figure shows three right triangles. triangles pqs, qrs, and prq are similar.

theorem: if two triangles are similar, the corresponding sides are in proportion.

--figure shows triangle pqr with right angle at q. segment pq is 4 and segment qr is 9. point s is on segment pr and angles qsp and qsr are right

using the given theorem, which two statements to prove that if segment pr is x, then x2 = 97?

a. segment pr • segment ps = 16
segment pr • segment sr = 36

b. segment pr • segment ps = 36
segment pr • segment sr = 81

c. segment pr • segment ps = 16
segment pr • segment sr = 81

d. segment pr • segment ps = 81
segment pr • segment sr = 16

3 right triangles:
ΔPQR ; long leg (QR) = 9 ; short leg (PQ) = 4 ; hypotenuse (PR) = ?
ΔQSP ; long leg (QS) = ? ; short leg (SP) = ? ; hypotenuse (PQ) = 4
ΔRSQ ; long leg (RS) = ? ; short leg (SQ) = ? ; hypotenuse (QR) = 9

Let us use the Pythagorean theorem to solve for the hypotenuse of ΔPQR
a² + b² = c²
9² + 4² = c²
81 + 16 = c²
97 = c²
√97 = √c²
9.849 = c  Hypotenuse (PR)

Compare ΔPQR and ΔQSP using hypotenuse and long leg
9.849/9 = 4/x 
9.848x = 9*4
9.848x = 36
x = 36/9.848
x = 3.656 long leg (QS) of ΔQSP 

QS is the long leg of ΔQSP and the short leg of ΔRSQ

ΔPQR ; long leg (QR) = 9 ; short leg (PQ) = 4 ; hypotenuse (PR) = 9.849
ΔQSP ; long leg (QS) = 3.656 ; short leg (SP) = ? ; hypotenuse (PQ) = 4
ΔRSQ ; long leg (RS) = ? ; short leg (SQ) = 3.656 ; hypotenuse (QR) = 9

To get the measure of the missing leg; we simply solve it using Pythagorean Theorem.
ΔQSP: 3.656² + b² = 4²
b² = 4² - 3.656²
b² = 16 - 13.366
b² = 2.634
√b² = √2.634
b = 1.623  short leg (SP)

ΔRSQ: a² + 3.656² = 9²
a² = 9² - 3.656²
a² = 81 - 13.366
a² = 67.634
√a² = √67.634
a = 8.224 long leg (RS)

ΔPQR ; long leg (QR) = 9 ; short leg (PQ) = 4 ; hypotenuse (PR) = 9.849
ΔQSP ; long leg (QS) = 3.656 ; short leg (SP) = 1.623 ; hypotenuse (PQ) = 4
ΔRSQ ; long leg (RS) = 8.224 ; short leg (SQ) = 3.656 ; hypotenuse (QR) = 9

My answer is: 
C. Segment PR • segment PS = 16   →→ 9.849 * 1.623 = 15.985 or 16
Segment PR • segment SR = 81 →→ 9.849 * 8.224 = 80.998 or 81

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