Shape and Space 2
5 marks · HardThe diagram below shows a tangent to the circle with centre O. P is the point of contact, and the radius OP is 8 cm. The tangent TP meets a straight line passing through the centre O at point T, such that angle OTP = 35°.
State the relationship between the tangent and the radius at point P.
Calculate the length TP. Give your answer to 3 significant figures.
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) The tangent is perpendicular to the radius at the point of contact.
(b) 11.4 cm
Also accept: Exact answer or correct working using trigonometry
Common mistakes: Not stating the tangent is perpendicular; Incorrect use of trigonometric ratios
Show mark scheme and worked solution
Marking points
B1 (a) · 1 mark · Correctly states the tangent is perpendicular to the radius at point P.
M1 (b) · 1 mark · Recognises triangle OPT is right-angled at P.
M2 (b) · 1 mark · Sets up trigonometric relationship (TP as opposite to 35°, OP as adjacent, uses tan(θ) = opp/adj).
A1 (b) · 1 mark · Works out TP = 8 * tan(35°).
A2 (b) · 1 mark · Gives final answer: TP = 5.60 cm (to 3 s.f.)
Worked solution
a) The tangent to a circle is perpendicular to the radius at the point of contact (P).
b) Triangle OPT is right-angled at P with OP = 8 cm and angle OTP = 35°.
To 3 significant figures: TP = 5.60 cm.