Theory question 102
Triangles and QuadrilateralsIn the diagram below, ∠PQR = ∠PSQ = 90°, [PS] = 9cm, [SR] = 16cm and [SQ] = xcm.
(a) Find the value of x using a trigonometric ratio.
(b) Calculate: (i) the size of QRS correct to the nearest degree, (ii) PQ.
Show the model answer
Model answer
(a) Let ∠QRS = θ. In △QSR: tan θ = x/16.
∠PQS = 90° − ∠SQR = θ, so in △PQS: tan θ = 9/x.
x/16 = 9/x ⇒ x2 = 144 ⇒ x = 12 cm.
(b)(i) tan ∠QRS = 12/16 = 0.75 ⇒ ∠QRS = 36.87° ≈ 37°.
(ii) PQ2 = PS2 + SQ2 = 81 + 144 = 225 ⇒ PQ = 15 cm (or PQ = PR sin 36.87° = 25 × 0.6 = 15 cm).
Explanation
Spot that ∠PQS equals ∠QRS; that links the two triangles.