Geometry Study Resources for Competitive Exams

Geometry

Q. If the coordinates of points A, B, and C are (1, 2), (4, 6), and (7, 2) respectively, what is the area of triangle ABC?

A. 12
B. 10
C. 14
D. 8

Solution

Area = 1/2 * |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| = 1/2 * |1(6-2) + 4(2-2) + 7(2-6)| = 1/2 * |4 + 0 - 28| = 12.

Correct Answer: B — 10

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Q. If the coordinates of the center of a circle are (0, 0) and it passes through the point (3, 4), what is the radius of the circle?

A. 3 units
B. 4 units
C. 5 units
D. 7 units

Solution

Radius = distance from center to point = √[(3 - 0)² + (4 - 0)²] = √[9 + 16] = √25 = 5 units.

Correct Answer: C — 5 units

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Q. If the coordinates of the center of a circle are (2, 3) and the radius is 4, what is the equation of the circle?

A. (x - 2)² + (y - 3)² = 16
B. (x + 2)² + (y + 3)² = 16
C. (x - 2)² + (y + 3)² = 16
D. (x + 2)² + (y - 3)² = 16

Solution

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Correct Answer: A — (x - 2)² + (y - 3)² = 16

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Q. If the coordinates of the center of a circle are (3, 4) and the radius is 5, what is the equation of the circle?

A. (x-3)² + (y-4)² = 25
B. (x+3)² + (y+4)² = 25
C. (x-3)² + (y-4)² = 5
D. (x-3)² + (y-4)² = 20

Solution

The standard form of the equation of a circle is (x-h)² + (y-k)² = r², where (h, k) is the center and r is the radius.

Correct Answer: A — (x-3)² + (y-4)² = 25

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Q. If the coordinates of the center of a circle are (4, -2) and it passes through the point (4, 2), what is the radius of the circle?

A. 4
B. 2
C. 6
D. 8

Solution

Using the distance formula: radius = √((4 - 4)² + (2 - (-2))²) = √(0 + 16) = √16 = 4.

Correct Answer: B — 2

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Q. If the coordinates of the vertices of a triangle are (0, 0), (4, 0), and (0, 3), what is the area of the triangle?

A. 6 square units
B. 12 square units
C. 8 square units
D. 10 square units

Solution

The area of a triangle can be calculated using the formula A = 1/2 * base * height. Here, base = 4 and height = 3, so A = 1/2 * 4 * 3 = 6 square units.

Correct Answer: A — 6 square units

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Q. If the coordinates of the vertices of a triangle are (0, 0), (4, 0), and (0, 3), what is the perimeter of the triangle?

A. 12
B. 10
C. 14
D. 16

Solution

The lengths of the sides are 4, 3, and 5 (using the Pythagorean theorem). Therefore, the perimeter = 4 + 3 + 5 = 12.

Correct Answer: A — 12

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Q. If the coordinates of the vertices of a triangle are (0,0), (4,0), and (0,3), what is its area?

A. 6 cm²
B. 12 cm²
C. 8 cm²
D. 10 cm²

Solution

The area of a triangle can be calculated using the formula A = 1/2 * base * height. Here, base = 4 cm and height = 3 cm, so A = 1/2 * 4 * 3 = 6 cm².

Correct Answer: A — 6 cm²

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Q. If the coordinates of the vertices of a triangle are (1, 1), (4, 1), and (1, 5), what is the length of the base?

A. 3
B. 4
C. 5
D. 6

Solution

The base is the distance between (1, 1) and (4, 1): d = √((4-1)² + (1-1)²) = √9 = 3.

Correct Answer: A — 3

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Q. If the coordinates of the vertices of a triangle are A(1, 1), B(4, 1), and C(1, 5), what is the perimeter of the triangle?

A. 10
B. 12
C. 14
D. 16

Solution

Length AB = 3, Length AC = 4, Length BC = √((4-1)² + (1-5)²) = √(9 + 16) = 5. Perimeter = 3 + 4 + 5 = 12.

Correct Answer: B — 12

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Q. If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the perimeter of the triangle?

A. 12
B. 14
C. 16
D. 18

Solution

Perimeter = AB + BC + CA = √((4-1)² + (5-1)²) + √((7-4)² + (2-5)²) + √((1-7)² + (1-2)²) = 3 + √(9 + 9) + √(36 + 1) = 3 + 4.24 + 6.08 = 13.32.

Correct Answer: B — 14

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Q. If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the length of side AB?

A. 5
B. 4
C. 3
D. 6

Solution

Length AB = √((4 - 1)² + (5 - 1)²) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. If the coordinates of the vertices of a triangle are A(1, 2), B(4, 6), and C(1, 6), what is the base of the triangle AB?

A. 3
B. 4
C. 5
D. 6

Solution

Base AB = √((4-1)² + (6-2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5.

Correct Answer: A — 3

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Q. If the coordinates of the vertices of a triangle are A(1, 2), B(4, 6), and C(1, 6), what is the area of the triangle?

A. 6 square units
B. 8 square units
C. 10 square units
D. 12 square units

Solution

Using the formula Area = 1/2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|, we find Area = 1/2 |1(6 - 6) + 4(6 - 2) + 1(2 - 6)| = 1/2 |0 + 16 - 4| = 1/2 * 12 = 6 square units.

Correct Answer: A — 6 square units

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Q. If the coordinates of the vertices of a triangle are A(1, 2), B(4, 6), and C(7, 2), what is the perimeter of the triangle?

A. 12
B. 14
C. 16
D. 18

Solution

Perimeter = AB + BC + CA = √((4-1)² + (6-2)²) + √((7-4)² + (2-6)²) + √((1-7)² + (2-2)²) = 3 + 5 + 6 = 14.

Correct Answer: B — 14

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Q. If the coordinates of the vertices of triangle ABC are A(1, 2), B(4, 6), and C(1, 6), what is the length of side AB?

A. 3
B. 4
C. 5
D. 6

Solution

Using the distance formula, AB = √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5.

Correct Answer: B — 4

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Q. If the coordinates of the vertices of triangle PQR are P(1, 2), Q(4, 6), and R(1, 6), what is the length of side PQ?

A. 3
B. 4
C. 5
D. 6

Solution

Using the distance formula, PQ = √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5.

Correct Answer: B — 4

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Q. If the diagonals of a rectangle are 10 cm long, what is the length of each side if the rectangle is a square?

A. 5 cm
B. 10 cm
C. 7.07 cm
D. 8 cm

Solution

In a square, the diagonal d is related to the side length s by the formula d = s√2. Therefore, s = d/√2 = 10 cm/√2 = 10 cm/1.414 ≈ 7.07 cm.

Correct Answer: C — 7.07 cm

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Q. If the diagonals of a rhombus are 10 cm and 24 cm, what is the area of the rhombus?

A. 120 cm²
B. 60 cm²
C. 80 cm²
D. 100 cm²

Solution

The area of a rhombus can be calculated using the formula: Area = 1/2 × d1 × d2, where d1 and d2 are the lengths of the diagonals. Therefore, Area = 1/2 × 10 cm × 24 cm = 120 cm².

Correct Answer: A — 120 cm²

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Q. If the diagonals of a rhombus are 10 cm and 24 cm, what is the area?

A. 120 cm²
B. 240 cm²
C. 300 cm²
D. 480 cm²

Solution

The area of a rhombus can be calculated using the formula (1/2) × d1 × d2, where d1 and d2 are the lengths of the diagonals. Here, area = (1/2) × 10 cm × 24 cm = 120 cm².

Correct Answer: A — 120 cm²

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Q. If the diagonals of a rhombus are 12 cm and 16 cm, what is the area of the rhombus?

A. 96 cm²
B. 48 cm²
C. 192 cm²
D. 64 cm²

Solution

The area of a rhombus can be calculated using the formula: Area = (d1 * d2) / 2. Here, d1 = 12 cm and d2 = 16 cm, so Area = (12 * 16) / 2 = 96 cm².

Correct Answer: A — 96 cm²

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Q. If the diagonals of a rhombus are 6 cm and 8 cm, what is its area?

A. 24 cm²
B. 30 cm²
C. 36 cm²
D. 48 cm²

Solution

The area of a rhombus is calculated as (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals. Thus, the area is (6 cm × 8 cm) / 2 = 24 cm².

Correct Answer: A — 24 cm²

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Q. If the diagonals of a rhombus are 6 cm and 8 cm, what is the area of the rhombus?

A. 24 cm²
B. 30 cm²
C. 48 cm²
D. 60 cm²

Solution

The area of a rhombus can be calculated using the formula (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals. Thus, the area is (6 cm * 8 cm) / 2 = 24 cm².

Correct Answer: A — 24 cm²

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Q. If the diagonals of a rhombus are 8 cm and 6 cm, what is the area?

A. 24 cm²
B. 48 cm²
C. 36 cm²
D. 30 cm²

Solution

The area of a rhombus can be calculated using the formula (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals. Here, area = (8 cm × 6 cm) / 2 = 24 cm².

Correct Answer: A — 24 cm²

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Q. If the diameter of a circle is 10 cm, what is the area of the circle?

A. 25π cm²
B. 50π cm²
C. 100π cm²
D. 75π cm²

Solution

The radius r = diameter/2 = 10/2 = 5 cm. The area A = πr² = π(5)² = 25π cm².

Correct Answer: A — 25π cm²

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Q. If the diameter of a circle is 10 cm, what is the circumference of the circle?

A. 31.4 cm
B. 20 cm
C. 15.7 cm
D. 25 cm

Solution

The circumference of a circle is given by the formula C = πd. For d = 10 cm, C = π(10) ≈ 31.4 cm.

Correct Answer: A — 31.4 cm

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Q. If the diameter of a circle is 10 cm, what is the circumference?

A. 31.4 cm
B. 25.0 cm
C. 15.7 cm
D. 20.0 cm

Solution

The circumference of a circle is given by C = πd. Here, C = π(10) ≈ 31.4 cm.

Correct Answer: A — 31.4 cm

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Q. If the diameter of a circle is 12 cm, what is its area?

A. 113.04 cm²
B. 36 cm²
C. 144 cm²
D. 28.26 cm²

Solution

Radius = diameter / 2 = 12 / 2 = 6 cm. Area = πr² = π(6)² = 36π ≈ 113.04 cm².

Correct Answer: A — 113.04 cm²

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Q. If the diameter of a circle is 12 cm, what is the area of the circle?

A. 36π cm²
B. 144π cm²
C. 72π cm²
D. 24π cm²

Solution

The radius r is half of the diameter, so r = 12/2 = 6 cm. The area A = πr² = π(6)² = 36π cm².

Correct Answer: C — 72π cm²

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Q. If the diameter of a circle is 14 cm, what is the area of the circle?

A. 154 cm²
B. 196 cm²
C. 100 cm²
D. 120 cm²

Solution

The radius r = diameter/2 = 14/2 = 7 cm. The area A = πr² = π(7)² = 49π ≈ 154 cm².

Correct Answer: B — 196 cm²

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Geometry

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