10th grade geometry

Example

ABCD is a cyclic quadrilateral. Angle A = 110°. What is angle C?

1. Opposite angles in a cyclic quadrilateral add up to 180°.
2. 70°

Answer: 70

Example

a = (3, −3) and b = (−6, 6). Find 3a + 3b. Type it like (2, −1).

1. 3a = (9, −9)
2. 3b = (−18, 18)
3. (−9, 9)

Answer: (−9, 9)

Example

A cuboid measures 7 cm × 10 cm × 2 cm. Find the length of its space diagonal, to 1 decimal place.

1. 7² + 10² + 2² = 153
2. √153 ≈ 12.4 cm

Answer: 12.4

Example

What is the centre of the circle (x + 6)² + (y − 5)² = 16? Type it like (2, −1).

1. (−6, 5)

Answer: (−6, 5)

Example

A line crosses two parallel lines. Angle p and an angle of 139° are between the parallel lines, on opposite sides of the crossing line (a Z shape). Which reason should go with the step that finds p?

1. Alternate angles are equal

Answer: Alternate angles are equal

Example

Triangles ABC and DEF have: ∠A = ∠D, AB = DE, ∠B = ∠E. Which test proves they are congruent?

1. ASA

Answer: ASA

Example

Triangle P has sides 5, 7, 8 cm. Triangle Q has sides 14.5, 17.5, 20 cm. Are they similar?

1. 14.5 ÷ 5 = 2.9; 17.5 ÷ 7 = 2.5; 20 ÷ 8 = 2.5
2. No

Answer: No

Example

Triangles ABC and PQR are similar, with AB matching PQ and BC matching QR. AB = 7 cm, PQ = 28 cm and BC = 10 cm. Find QR.

1. k = 28 ÷ 7 = 4
2. QR = 10 × 4 = 40 cm

Answer: 40

Example

Two similar shapes have lengths in the ratio 1 : 3. The smaller has area 14 cm². What is the area of the larger?

1. 14 × 3² = 14 × 9 = 126 cm²

Answer: 126

Example

The angles of a triangle are (x − 6)°, (3x)° and (x + 11)°. Find x.

1. 5x + 5 = 180
2. x = 35

Answer: 35

Example

A triangle has interior angles 26° and 76°. One side is extended at the third vertex. What is the exterior angle there?

1. 26° + 76° = 102°

Answer: 102

Example

An isosceles triangle has an angle of 64° between its two equal sides. Find one of the other angles.

1. (180° − 64°) ÷ 2 = 58°

Answer: 58

Example

P divides AB so that AP : PB = 2 : 1. A is (−5, 1) and P is (1, 7). Find B. Type it like (2, −1).

1. one part = (3, 3)
2. (4, 10)

Answer: (4, 10)

Example

Find the exact distance between (5, 3) and (6, −4). Simplify the surd.

1. d = √(1² + 7²) = √50
2. = 5√2

Answer: 5√2

Example

Which is the equation of the circle with centre (−1, −1) and radius 4?

1. (x + 1)² + (y + 1)² = 16

Answer: (x + 1)² + (y + 1)² = 16

Example

Find the centre of the circle x² + y² + 6x − 12y + 36 = 0. Type it like (2, −1).

1. (x + 3)² + (y − 6)² = 9
2. (−3, 6)

Answer: (−3, 6)

Example

O is the centre of a circle. A, B and C are on the circle, with C on the major arc. ∠ACB = 49°. Find ∠AOB.

1. ∠AOB = 2 × 49° = 98°

Answer: 98

Example

ABCD is a cyclic quadrilateral. ∠DAB = 128°. Find ∠BCD.

1. 180° − 128° = 52°

Answer: 52

Example

PA and PB are tangents from P to a circle with centre O, touching it at A and B. ∠APB = 49°. Find ∠AOB.

1. ∠OAP = ∠OBP = 90°
2. ∠AOB = 360° − 90° − 90° − 49° = 131°

Answer: 131

Example

A tangent touches a circle at A. Chord AB makes an angle of 61° with the tangent. C is a point on the circle in the other segment. Find ∠ACB.

1. ∠ACB = 61°

Answer: 61

Example

O is the centre of the circle. Which theorem justifies this step? ∠OTP = 90°, where PT touches the circle at T

1. A tangent is perpendicular to the radius

Answer: A tangent is perpendicular to the radius

Example

A sector of a circle with radius 3 cm has an arc length of 3π cm. Find the angle of the sector in degrees.

1. θ/360 = 3π ÷ 6π
2. θ = 180°

Answer: 180

Example

Find the perimeter of a sector with radius 13 cm and angle 210°. Give your answer to 1 decimal place.

1. arc = 210/360 × 2π × 13 ≈ 47.647
2. P = 47.647 + 26 ≈ 73.6 cm

Answer: 73.6

Example

A rocket model is a cylinder (radius 5 cm, height 9 cm) with a cone (same radius, height 9 cm) on top. Find its volume to 1 decimal place.

1. π × 5² × 9 ≈ 706.86
2. ⅓ × π × 5² × 9 ≈ 235.62
3. ≈ 942.5 cm³

Answer: 942.5

Example

Find the surface area of a sphere of radius 12 cm. Give an exact answer in terms of π.

1. 4 × π × 12² = 576π cm²

Answer: 576*pi

Example

A cone has base radius 18 cm and height 12 cm. The top is cut off parallel to the base, leaving a small cone of radius 12 cm and height 8 cm removed. Find the volume of the frustum that is left, to 1 decimal place.

1. ⅓π(18² × 12 − 12² × 8)
2. ≈ 2865.1 cm³

Answer: 2865.1

Example

P is (5, −2) and the vector PQ is (−1, 5). Find the coordinates of Q. Type them like (2, −1).

1. (4, 3)

Answer: (4, 3)

Example

Find the magnitude of (7, 6) to 1 decimal place.

1. √(49 + 36) = √85
2. ≈ 9.2

Answer: 9.2

Example

OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.

1. OM = OA + ½AB = a + ½(b − a)
2. ½a + ½b

Answer: ½a + ½b

Example

Is 8a − 4b parallel to −4a + 3b?

1. 8 ÷ (−4) = −2; −4 ÷ 3 = −1.333
2. No

Answer: No