Mathematics Test Questions
5C Problems involving triangles
cQ1. The diagram shows a sector AOB of a circle of radius 15 cm and centre O. The angle at the centre of the circle is 115.
Calculate (a) the area of the sector AOB.
(b) the area of the shaded region. (226 , 124
nQ2. Consider a triangle and two arcs of circles.
The triangle ABC is a right-angled isosceles triangle, with AB = AC = 2.
The point P is the midpoint of [BC].
The arc BDC is part of a circle with centre A.
The arc BEC is part of a circle with centre P.
(a) Calculate the area of the segment BDCP. (b) Calculate the area of the shaded region BECD.
cQ3. In the following diagram, O is the centre of the circle and (AT) is the tangent to the circle at T. If OA = 12 cm, and the circle has a radius of 6 cm, find the area of the shaded region.
cQ4. The diagram shows a circle, centre O, with a radius 12 cm. The chord AB subtends at an angle of 75° at the centre. The tangents to the circle at A and B meet at P.
(a) Using the cosine rule, show that the length of AB is (b) Find the length of BP. (c) Hence find
(i) the area of triangle OBP;
(ii) the area of triangle ABP. (d) Find the area of sector OAB.
(e) Find the area of the shaded region.
Miscellaneous Problems
Q5. The diagram below shows a circle with centre O and radius 8 cm. The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length of arc ABC is 6 cm.
(a) Find the size of angle AOC.
(b) Hence find the area of the shaded region. The area of sector OCDE is 45 cm2.
(c) Find the size of angle COE.
(d) Find EF.
Q6. The following diagram shows a sector of a circle of radius r cm, and angle at the centre. The perimeter of the sector is 20 cm.
(a) Show that = .
(b) The area of the sector is 25 cm2. Find the value of r.
Q7. The following diagram shows
cQ1. The diagram shows a sector AOB of a circle of radius 15 cm and centre O. The angle at the centre of the circle is 115.
Calculate (a) the area of the sector AOB.
(b) the area of the shaded region. (226 , 124
nQ2. Consider a triangle and two arcs of circles.
The triangle ABC is a right-angled isosceles triangle, with AB = AC = 2.
The point P is the midpoint of [BC].
The arc BDC is part of a circle with centre A.
The arc BEC is part of a circle with centre P.
(a) Calculate the area of the segment BDCP. (b) Calculate the area of the shaded region BECD.
cQ3. In the following diagram, O is the centre of the circle and (AT) is the tangent to the circle at T. If OA = 12 cm, and the circle has a radius of 6 cm, find the area of the shaded region.
cQ4. The diagram shows a circle, centre O, with a radius 12 cm. The chord AB subtends at an angle of 75° at the centre. The tangents to the circle at A and B meet at P.
(a) Using the cosine rule, show that the length of AB is (b) Find the length of BP. (c) Hence find
(i) the area of triangle OBP;
(ii) the area of triangle ABP. (d) Find the area of sector OAB.
(e) Find the area of the shaded region.
Miscellaneous Problems
Q5. The diagram below shows a circle with centre O and radius 8 cm. The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length of arc ABC is 6 cm.
(a) Find the size of angle AOC.
(b) Hence find the area of the shaded region. The area of sector OCDE is 45 cm2.
(c) Find the size of angle COE.
(d) Find EF.
Q6. The following diagram shows a sector of a circle of radius r cm, and angle at the centre. The perimeter of the sector is 20 cm.
(a) Show that = .
(b) The area of the sector is 25 cm2. Find the value of r.
Q7. The following diagram shows