In the given triangle PQR, angle P is 90° and segment PS is perpendicular to segment QR. Q
S
R
P
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)

Step-by-step explanation:

A) The pair of similar triangles are both Isosceles triangle or right angled triangles.

B) An isosceles triangle is a triangle that has two of its sides and angle equal while a right angled triangle has one of its angles to e 90°. Since he line PS cuts RS at Q and perpendicular to the line RS, then ∠PSR = ∠PSQ = 90°. This shows that one of the angles of ΔPSQ and ΔPSR is 90° proving that they are both right angled triangle.

Similarly, it can be seen that ∠RPQ is a right angled triangled divided equally into ∠SPQ and ∠SPR. This shows that  ∠SPQ =  ∠SPR = 45°.

Note that ∠PSR = ∠PSQ = 90°,

From ΔPSQ, ∠SPQ.+∠SQP.+∠PSQ= 180°,

45+∠SQP+90 = 180

∠SQP = 180-135

∠SQP = 45

Since ∠SQP = ∠SPQ in ΔPSQ, this shows that the triangle is an isosceles triangle.

Also since ∠SPR = ∠SRP = 45 in ΔSPR, this shows that the triangle is an isosceles triangle.

C) RS = 4 and RQ = 16, find the length of segment RP

From ΔSRP, sin ∠SPR = RS/RP

Since ∠SPR = 45°

sin 45° = 4/RP

RP = 4/sin45°

RP = 4/(1/√2)

RP = 4√2

step-by-step explanation:

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