Geometry Conceptual Quiz Set 1 - Concept of Maths | Concept of Maths

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 $12\text{ cm}^2$, calculate the total area of $\triangle ABC$.

A) $24\text{ cm}^2$

B) $30\text{ cm}^2$

C) $36\text{ cm}^2$

D) $48\text{ cm}^2$

Answer

Correct Answer: C

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

Question 2

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

A) $49\text{ cm}^2$

B) $98\text{ cm}^2$

C) $72\text{ cm}^2$

D) $196\text{ cm}^2$

Answer

Correct Answer: B

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

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) 9

B) 12

C) 14

D) 18

Answer

Correct Answer: A

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: D

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 $24\text{ cm}$ is located $5\text{ cm}$ from the center. What is the length of another chord in the same circle that is $12\text{ cm}$ from the center?

A) $5\text{ cm}$

B) $10\text{ cm}$

C) $13\text{ cm}$

D) $15\text{ cm}$

Answer

Correct Answer: B

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

Question 6

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

A) $2.5\text{ cm}$

B) $3.5\text{ cm}$

C) $3.0\text{ cm}$

D) $4.2\text{ cm}$

Answer

Correct Answer: A

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 $BC = 12k = 6\text{ cm}$. $k = \frac{6}{12} = 0.5$. $BD = 5 \times 0.5 = 2.5\text{ cm}$.

Question 7

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

A) $18\text{ cm}$

B) $9\text{ cm}$

C) $8\text{ cm}$

D) $6.5\text{ cm}$

Answer

Correct Answer: C

When two circles touch internally, the distance between their centers is the difference between their radii. Distance $= |R - r| = |13 - 5| = 8\text{ cm}$.

Question 8

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

A) $24\text{ cm}$

B) $36\text{ cm}$

C) $48\text{ cm}$

D) $12\sqrt{3}\text{ cm}$

Answer

Correct Answer: B

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 $a = 12\text{ cm}$. Perimeter $= 3a = 3 \times 12 = 36\text{ cm}$.

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) 30

B) 35

C) 45

D) 40

Answer

Correct Answer: D

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 $13\text{ cm}$ and $15\text{ cm}$ intersect each other, and the length of their common chord is $24\text{ cm}$. Calculate the distance between their centers?

A) $12\text{ cm}$

B) $13\text{ cm}$

C) $14\text{ cm}$

D) $16\text{ cm}$

Answer

Correct Answer: C

The common chord is bisected perpendicularly by the line joining the centers. Half the chord $= 12\text{ cm}$. 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 $= d_1 + d_2 = 5 + 9 = 14\text{ cm}$.

Competitive Exams : Quantitative Aptitude - Geometry (Set 1) | Concept of Maths

Competitive Exams : Quantitative Aptitude - Geometry (Set 1) | Concept of Maths

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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 $24\text{ cm}$ is located $5\text{ cm}$ from the center. What is the length of another chord in the same circle that is $12\text{ cm}$ from the center?

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

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

8. The area of an equilateral triangle is $36\sqrt{3}\text{ cm}^2$. 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 $13\text{ cm}$ and $15\text{ cm}$ intersect each other, and the length of their common chord is $24\text{ cm}$. Calculate the distance between their centers?

Geometry Conceptual Quiz Set 1 - Concept of Maths | Concept of Maths

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 $12\text{ cm}^2$, calculate the total area of $\triangle ABC$.

2. A circle is inscribed inside a square of side length $14\text{ cm}$. 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.

Competitive Exams : Quantitative Aptitude - Geometry (Set 1) | Concept of Maths

Competitive Exams : Quantitative Aptitude - Geometry (Set 1) | Concept of Maths

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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 $24\text{ cm}$ is located $5\text{ cm}$ from the center. What is the length of another chord in the same circle that is $12\text{ cm}$ from the center?

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

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

8. The area of an equilateral triangle is $36\sqrt{3}\text{ cm}^2$. 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 $13\text{ cm}$ and $15\text{ cm}$ intersect each other, and the length of their common chord is $24\text{ cm}$. Calculate the distance between their centers?