Shape and Space 3
5 marks · HardThe diagram below shows triangle PQR. PQ = 9.0 cm, PR = 6.2 cm and angle QPR = 48°.
Calculate the length QR, giving your answer correct to 3 significant figures.
Write down which side in triangle PQR is the hypotenuse relative to angle QPR.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) 8.5 cm
(b) QR
Common mistakes: Mistaking sides in relation to given angle; Incorrect law or identity
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Uses correct trigonometric relation (cos or sin) for the triangle.
M2 (a) · 1 mark · Uses correct values for substitution. (follow-through)
A1 (a) · 1 mark · Obtains value for required side using correct working. (follow-through)
A2 (a) · 1 mark · Rounds value to 3 significant figures. (follow-through)
B1 (b) · 1 mark · Identifies QR as the hypotenuse relative to QPR.
Worked solution
Label triangle with PQ = 9.0 cm, PR = 6.2 cm, angle QPR = 48°.
Since we have 2 sides and the included angle, use the cosine rule:
Calculate: 2 x 9.0 x 6.2 x 0.6691 ≈ 74.88
But this does not match a right-angled triangle. If triangle is right-angled at QPR, then PQ and PR are legs, and QR is the hypotenuse.
By Pythagoras: QR^2 = PQ^2 + PR^2 = 9.0^2 + 6.2^2 = 81 + 38.44 = 119.44 → QR = 10.93.
But angle QPR ≠ 90°, must use Law of Cosines (above).
To 3 significant figures: QR ≈ 8.54 cm.
QR is the side opposite to angle QPR, and with available data and this angle, QR is the longest side: the hypotenuse with respect to QPR.