Geometry Quiz: Triangles, Circles and Polygons Practice

Spoilers! These are the Starter quiz’s answers. Easy, Medium, Hard and Mixed rounds show their own answers when you finish.

1. The angles of a triangle are in the ratio 2:3:4. What is the largest angle?

   • 60°
   • 80°
   • 90°
   • 72°

   Answer: 80°. The angles add to 180°, so one part = 180/9 = 20°. Largest = 4 × 20 = 80°.

2. Each interior angle of a regular polygon is 140°. How many sides does it have?

   • 8
   • 10
   • 9
   • 12

   Answer: 9. Exterior angle = 180 − 140 = 40°, and the exterior angles add to 360°, so n = 360/40 = 9.

3. How many diagonals does a decagon (10-sided polygon) have?

   • 35
   • 45
   • 40
   • 30

   Answer: 35. Diagonals = n(n − 3)/2 = 10 × 7/2 = 35.

4. A point P is 13 cm from the centre of a circle of radius 5 cm. What is the length of the tangent drawn from P to the circle?

   • 8 cm
   • 10 cm
   • 11 cm
   • 12 cm

   Answer: 12 cm. The radius meets the tangent at 90°, so tangent = sqrt(13² − 5²) = sqrt(144) = 12 cm.

5. An arc of a circle subtends an angle of 110° at the centre. What angle does it subtend at any point on the remaining part of the circle?

   • 110°
   • 55°
   • 70°
   • 125°

   Answer: 55°. The angle at the circumference is half the angle at the centre on the same arc: 110/2 = 55°.

6. The areas of two similar triangles are in the ratio 16:25. What is the ratio of their corresponding sides?

   • 16:25
   • 256:625
   • 4:5
   • 2:3

   Answer: 4:5. Area ratio = (side ratio)², so side ratio = sqrt(16):sqrt(25) = 4:5.

7. In triangle ABC, AD is a median and G is the centroid. If AG = 12 cm, what is the length of AD?

   • 18 cm
   • 24 cm
   • 16 cm
   • 20 cm

   Answer: 18 cm. The centroid divides each median 2:1 from the vertex, so AG = 2/3 of AD: AD = 12 × 3/2 = 18 cm.

8. What is the radius of the circle inscribed in a right triangle whose perpendicular sides are 6 cm and 8 cm?

   • 3 cm
   • 4 cm
   • 2.5 cm
   • 2 cm

   Answer: 2 cm. Hypotenuse = 10. For a right triangle, r = (a + b − c)/2 = (6 + 8 − 10)/2 = 2 cm. (Area 24 ÷ semi-perimeter 12 also gives 2.)

9. ABCD is a cyclic quadrilateral in which angle A = 75°. What is angle C?

   • 75°
   • 95°
   • 105°
   • 115°

   Answer: 105°. Opposite angles of a cyclic quadrilateral add up to 180°, so C = 180 − 75 = 105°.

10. Which of these is NOT a valid rule for proving two triangles congruent?

    • SSA
    • SAS
    • ASA
    • RHS

    Answer: SSA. SSA (two sides and a non-included angle) can fit two different triangles. SSS, SAS, ASA, AAS and RHS are the valid rules.