Geometry Study Resources for Competitive Exams

Geometry

Q. If a parallelogram has vertices at (0, 0), (2, 3), (5, 3), and (3, 0), what is its area?

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

Solution

The area of a parallelogram can be calculated using the formula: Area = base * height. The base is 5 - 0 = 5 and the height is 3. Therefore, area = 5 * 3 = 15.

Correct Answer: B — 8

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Q. If a parallelogram has vertices at (2, 3), (5, 3), (6, 1), and (3, 1), what is its area?

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

Solution

Base = 5 - 2 = 3, Height = 3 - 1 = 2. Area = base * height = 3 * 2 = 6.

Correct Answer: B — 8

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Q. If a point P divides the line segment joining A(2, 3) and B(8, 7) in the ratio 1:3, what are the coordinates of point P?

A. (5, 5)
B. (4, 4)
C. (6, 6)
D. (3, 3)

Solution

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=1, n=3, A(2,3), B(8,7). P = ((1*8 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (5, 5).

Correct Answer: A — (5, 5)

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Q. If a point P divides the line segment joining A(2, 3) and B(8, 7) in the ratio 3:1, what are the coordinates of P?

A. (5, 4)
B. (6, 5)
C. (7, 6)
D. (4, 5)

Solution

Using the section formula: P = ((3*8 + 1*2)/(3+1), (3*7 + 1*3)/(3+1)) = (6, 5).

Correct Answer: B — (6, 5)

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Q. If a point P divides the segment joining A(1, 2) and B(5, 6) in the ratio 3:1, what are the coordinates of P?

A. (3, 4)
B. (4, 5)
C. (2.5, 3.5)
D. (3.5, 4.5)

Solution

Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) = ((3*5 + 1*1)/(3+1), (3*6 + 1*2)/(3+1)) = (3.5, 4.5).

Correct Answer: D — (3.5, 4.5)

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Q. If a point P divides the segment joining A(2, 1) and B(8, 5) in the ratio 3:1, what are the coordinates of P?

A. (5, 3)
B. (6, 4)
C. (4, 2)
D. (3, 2)

Solution

Using the section formula: P = ((3*8 + 1*2)/(3+1), (3*5 + 1*1)/(3+1)) = (5, 3).

Correct Answer: A — (5, 3)

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Q. If a point P divides the segment joining A(2, 3) and B(8, 7) in the ratio 3:1, what are the coordinates of P?

A. (5, 4)
B. (6, 5)
C. (7, 6)
D. (4, 5)

Solution

Using the section formula: P = ((3*8 + 1*2)/(3+1), (3*7 + 1*3)/(3+1)) = (6, 5).

Correct Answer: B — (6, 5)

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Q. If a polygon has 5 sides, what is it called?

A. Triangle
B. Quadrilateral
C. Pentagon
D. Hexagon

Solution

A polygon with 5 sides is called a pentagon.

Correct Answer: C — Pentagon

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Q. If a polygon has 8 sides, what is it called?

A. Hexagon
B. Heptagon
C. Octagon
D. Nonagon

Solution

A polygon with 8 sides is called an octagon.

Correct Answer: C — Octagon

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Q. If a quadrilateral has angles measuring 90 degrees, 85 degrees, 95 degrees, and x degrees, what is the value of x?

A. 90 degrees
B. 80 degrees
C. 85 degrees
D. 75 degrees

Solution

The sum of the interior angles of a quadrilateral is 360 degrees. Therefore, x = 360 - (90 + 85 + 95) = 360 - 270 = 90 degrees.

Correct Answer: D — 75 degrees

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Q. If a quadrilateral has sides of lengths 5 cm, 12 cm, 13 cm, and 14 cm, is it a cyclic quadrilateral?

A. Yes
B. No
C. Not enough information
D. Only if it is a rectangle

Solution

A quadrilateral is cyclic if the sum of opposite angles is 180 degrees. The given side lengths do not provide enough information to determine if it is cyclic.

Correct Answer: B — No

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Q. If a quadrilateral has two pairs of opposite sides that are equal, what can be concluded about the quadrilateral?

A. It is a rectangle
B. It is a rhombus
C. It is a parallelogram
D. It is a trapezoid

Solution

A quadrilateral with two pairs of opposite sides that are equal is a parallelogram by definition.

Correct Answer: C — It is a parallelogram

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Q. If a quadrilateral has two pairs of opposite sides that are equal, what type of quadrilateral is it?

A. Rectangle
B. Rhombus
C. Trapezoid
D. Parallelogram

Solution

A quadrilateral with two pairs of opposite sides that are equal is classified as a parallelogram.

Correct Answer: D — Parallelogram

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Q. If a quadrilateral is a kite, what can be said about its diagonals?

A. They are equal
B. They bisect each other
C. One diagonal bisects the other
D. They are perpendicular

Solution

In a kite, one diagonal bisects the other diagonal.

Correct Answer: C — One diagonal bisects the other

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Q. If a quadrilateral is a rectangle, what can be said about its diagonals?

A. They are equal and bisect each other
B. They are unequal
C. They are perpendicular
D. They are parallel

Solution

In a rectangle, the diagonals are equal in length and bisect each other.

Correct Answer: A — They are equal and bisect each other

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Q. If a quadrilateral is a rectangle, which of the following statements is true?

A. All sides are equal
B. Opposite sides are equal
C. All angles are acute
D. Diagonals are perpendicular

Solution

In a rectangle, opposite sides are equal in length.

Correct Answer: B — Opposite sides are equal

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Q. If a rectangle has a perimeter of 30 cm and a length of 10 cm, what is its width?

A. 5 cm
B. 7.5 cm
C. 10 cm
D. 12.5 cm

Solution

Perimeter = 2(length + width), so 30 cm = 2(10 cm + width). Width = (30 cm/2) - 10 cm = 5 cm.

Correct Answer: B — 7.5 cm

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Q. If a rectangle has vertices at (1, 1), (1, 4), (5, 1), and (5, 4), what is its area?

A. 12
B. 16
C. 20
D. 24

Solution

The area of a rectangle is given by the formula: Area = length * width. The length is 5 - 1 = 4 and the width is 4 - 1 = 3. Thus, Area = 4 * 3 = 12.

Correct Answer: B — 16

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Q. If a rectangle's length is doubled and its width is halved, what happens to its area?

A. It remains the same
B. It doubles
C. It halves
D. It quadruples

Solution

New area = (2 * length) * (width/2) = length * width = original area, so it doubles.

Correct Answer: B — It doubles

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Q. If a regular hexagon has a side length of 3 cm, what is the perimeter of the hexagon?

A. 9 cm
B. 12 cm
C. 15 cm
D. 18 cm

Solution

The perimeter of a regular hexagon is calculated as 6 times the side length. Therefore, perimeter = 6 × 3 cm = 18 cm.

Correct Answer: D — 18 cm

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Q. If a regular hexagon has a side length of 6 cm, what is its perimeter?

A. 24 cm
B. 30 cm
C. 36 cm
D. 42 cm

Solution

The perimeter of a regular hexagon is calculated as P = 6 * side length. Here, P = 6 * 6 cm = 36 cm.

Correct Answer: C — 36 cm

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Q. If a regular hexagon has a side length of 6 cm, what is the perimeter of the hexagon?

A. 24 cm
B. 30 cm
C. 36 cm
D. 42 cm

Solution

The perimeter of a regular hexagon is calculated as 6 times the side length. Therefore, perimeter = 6 × 6 cm = 36 cm.

Correct Answer: C — 36 cm

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Q. If a rhombus has diagonals of lengths 10 and 24, what is its area?

A. 120
B. 140
C. 160
D. 180

Solution

Area = (d1 * d2) / 2 = (10 * 24) / 2 = 120.

Correct Answer: A — 120

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Q. If a square has a perimeter of 32 cm, what is the area of the square?

A. 64 cm²
B. 128 cm²
C. 16 cm²
D. 32 cm²

Solution

Side length = perimeter/4 = 32/4 = 8 cm. Area = side² = 8² = 64 cm².

Correct Answer: A — 64 cm²

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Q. If a square has a perimeter of 40 cm, what is the area of the square?

A. 100 cm²
B. 160 cm²
C. 200 cm²
D. 250 cm²

Solution

Side length = perimeter/4 = 40/4 = 10 cm. Area = side² = 10² = 100 cm².

Correct Answer: A — 100 cm²

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Q. If a square has a perimeter of 40 cm, what is the length of one side?

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Solution

The perimeter of a square is calculated as 4 times the side length. Therefore, if the perimeter is 40 cm, then side length = 40 cm / 4 = 10 cm.

Correct Answer: B — 10 cm

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Q. If a square has a side length of 4 cm, what is its area?

A. 16 cm²
B. 12 cm²
C. 8 cm²
D. 20 cm²

Solution

The area of a square is given by the formula: Area = side². Here, side = 4 cm. Area = 4² = 16 cm².

Correct Answer: A — 16 cm²

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Q. If a square has a side length of 5 cm, what is its area?

A. 20 cm²
B. 25 cm²
C. 30 cm²
D. 15 cm²

Solution

The area of a square is given by the formula: Area = side². Here, Area = 5² = 25 cm².

Correct Answer: B — 25 cm²

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Q. If a tangent and a chord intersect at a point on the circle, what is the relationship between the angle formed and the angle subtended by the chord at the opposite arc?

A. They are equal
B. The tangent angle is double
C. The chord angle is double
D. They are supplementary

Solution

The angle formed by the tangent and the chord is equal to the angle subtended by the chord at the opposite arc.

Correct Answer: A — They are equal

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Q. If a tangent and a radius meet at a point on the circle, what is the angle between them?

A. 90 degrees
B. 45 degrees
C. 180 degrees
D. 0 degrees

Solution

The tangent to a circle at any point is perpendicular to the radius at that point, forming a 90-degree angle.

Correct Answer: A — 90 degrees

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Geometry

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