Question 1

1. In \triangle ABC, the point G represents the centroid. If the area of the smaller triangle formed by the centroid and the base BC (i.e., \triangle GBC) is $12cm^2$, calculate the total area of \triangle ABC.

A) $24cm^2$B) $30cm^2$C) $36cm^2$D) $48cm^2$

Answer

✓ Correct Answer: Option C

Explanation: The centroid of a triangle divides it into three triangles of equal area: \triangle GAB, \triangle GBC, and \triangle GCA. Given Area$(\triangle GBC) = 12cm^2$, the total area of $\triangle ABC = 3 \times Area(\triangle GBC) = 3 \times 12 = 36cm^2$.

Question 2

2. A circle is inscribed inside a square of side length 14 cm. Subsequently, a smaller square is inscribed inside this circle. What is the area of the innermost square?

A) $49cm^2$ $98cm^2$

Question 3

3. In a regular polygon, the ratio between an interior angle and an exterior angle is 7:2. Determine the number of sides of this polygon.

A) 9B) 12C) 14D) 18

Answer

✓ Correct Answer: Option A

Explanation: Let the interior angle be 7x and the exterior angle be 2x. We know, Interior Angle + Exterior Angle $= 180^\circ$. $7x + 2x = 180^\circ \Rightarrow 9x = 180^\circ \Rightarrow x = 20^\circ$. Exterior angle $= 2x = 40^\circ$. Number of sides $n = \frac{360^\circ}{Exterior Angle} = \frac{360}{40} = 9$.

Question 4

4. PQ is a chord of a circle with center O. A line PR is tangent to the circle at point P. If $\angle RPQ = 50^\circ$, find the measure of the central angle \angle POQ.

A) $50^\circ$B) $75^\circ$C) $80^\circ$D) $100^\circ$

Answer

✓ Correct Answer: Option D

Explanation: By the **Alternate Segment Theorem**, the angle between a tangent and a chord (\angle RPQ) is equal to the angle subtended by the chord in the alternate segment. Let A be a point on the major arc. Then $\angle PAQ = \angle RPQ = 50^\circ$. The angle subtended by a chord at the center is double the angle subtended at the circumference. $\angle POQ = 2 \times \angle PAQ = 2 \times 50^\circ = 100^\circ$.

Question 5

5. In a circle, a chord of length 24 cm is located 5 cm from the center. What is the length of another chord in the same circle that is 12 cm from the center?

A) 5 cmB) 10 cmC) 13 cmD) 15 cm

Answer

✓ Correct Answer: Option B

Explanation: First, find the radius r of the circle using the first chord. Radius $r^2 = (half chord)^2 + (distance)^2 = 12^2 + 5^2 = 144 + 25 = 169$. So r = 13. For the second chord at distance d = 12: $(half chord x)^2 = r^2 - d^2 = 13^2 - 12^2 = 169 - 144 = 25$. $x = \sqrt{25} = 5$. Total length of the second chord = 2x = 10cm.

Question 6

6. In \triangle ABC, the segment AD is the internal bisector of \angle BAC. If AB = 10cm, AC = 14cm, and BC = 6cm, find the length of the segment BD.

A) 2.5 cmB) 3.5 cmC) 3.0 cmD) 4.2 cm

Answer

✓ Correct Answer: Option A

Explanation: According to the **Angle Bisector Theorem**, the bisector of an angle of a triangle divides the opposite side into segments proportional to the other two sides. $\frac{BD}{DC} = \frac{AB}{AC} = \frac{10}{14} = \frac{5}{7}$. Let BD = 5k and DC = 7k. Then BC = 12k = 6cm. $k = \frac{6}{12} = 0.5$. $BD = 5 \times 0.5 = 2.5cm$.

Question 7

7. Two circles with radii 13 cm and 5 cm touch each other internally. What is the distance between their centers?

A) 18 cmB) 9 cmC) 8 cmD) 6.5 cm

Answer

✓ Correct Answer: Option C

Explanation: When two circles touch internally, the distance between their centers is the difference between their radii. Distance = |R - r| = |13 - 5| = 8cm.

Question 8

8. The area of an equilateral triangle is $36\sqrt{3}cm^2$. What is the perimeter of this triangle?

A) 24 cmB) 36 cmC) 48 cmD) $12\sqrt{3}cm$

Answer

✓ Correct Answer: Option B

Explanation: Area of equilateral triangle $= \frac{\sqrt{3}}{4} a^2$. $\frac{\sqrt{3}}{4} a^2 = 36\sqrt{3} \Rightarrow a^2 = 36 \times 4 = 144$. Side a = 12cm. Perimeter $= 3a = 3 \times 12 = 36cm$.

Question 9

9. In a cyclic quadrilateral ABCD, opposite angles \angle A and \angle C are given as $(2x+4)^\circ$ and $(4x-64)^\circ$ respectively. Find the value of x.

A) 30B) 35C) 45D) 40

Answer

✓ Correct Answer: Option D

Explanation: In a cyclic quadrilateral, the sum of opposite angles is $180^\circ$. $\angle A + \angle C = 180^\circ$ (2x + 4) + (4x - 64) = 180 6x - 60 = 180 \Rightarrow 6x = 240 \Rightarrow x = 40.

Question 10

10. Two circles of radii 13 cm and 15 cm intersect each other, and the length of their common chord is 24 cm. Calculate the distance between their centers?

A) 12 cmB) 13 cmC) 14 cmD) 16 cm

Answer

✓ Correct Answer: Option C

Explanation: The common chord is bisected perpendicularly by the line joining the centers. Half the chord = 12cm. Distance from first center $d_1 = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = 5$. Distance from second center $d_2 = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = 9$. Distance between centers $= d_1 + d_2 = 5 + 9 = 14cm$.

Accepted Answer

✓ Correct Answer: Option B

Explanation: 1. The diameter of the circle is equal to the side of the outer square = 14cm. 2. The diagonal of the inner square is equal to the diameter of the circle = 14cm. 3. Area of a square given its diagonal d is $\frac{1}{2} d^2$. 4. Area $= \frac{1}{2} \times 14^2 = \frac{1}{2} \times 196 = 98cm^2$.

Geometry Conceptual Quiz Set 1

1. In \triangle ABC, the point G represents the centroid. If the area of the smaller triangle formed by the centroid and the base BC (i.e., \triangle GBC) is $12cm^2$, calculate the total area of \triangle ABC.

2. A circle is inscribed inside a square of side length 14 cm. Subsequently, a smaller square is inscribed inside this circle. What is the area of the innermost square?

3. In a regular polygon, the ratio between an interior angle and an exterior angle is 7:2. Determine the number of sides of this polygon.

4. PQ is a chord of a circle with center O. A line PR is tangent to the circle at point P. If $\angle RPQ = 50^\circ$, find the measure of the central angle \angle POQ.

5. In a circle, a chord of length 24 cm is located 5 cm from the center. What is the length of another chord in the same circle that is 12 cm from the center?

6. In \triangle ABC, the segment AD is the internal bisector of \angle BAC. If AB = 10cm, AC = 14cm, and BC = 6cm, find the length of the segment BD.

7. Two circles with radii 13 cm and 5 cm touch each other internally. What is the distance between their centers?

8. The area of an equilateral triangle is $36\sqrt{3}cm^2$. What is the perimeter of this triangle?

9. In a cyclic quadrilateral ABCD, opposite angles \angle A and \angle C are given as $(2x+4)^\circ$ and $(4x-64)^\circ$ respectively. Find the value of x.

10. Two circles of radii 13 cm and 15 cm intersect each other, and the length of their common chord is 24 cm. Calculate the distance between their centers?