Geometry (Lines, Angles, Triangles, Polygons) Questions and Answers

Q: The number of sides of two regular polygons are in the ratio 5:4 and the difference between their interior angles is 9°. Find the number of sides.

Correct answer: 10, 8Let the polygons have \(5x\) and \(4x\) sides. Each interior angle of a regular \(n\)-gon is\[\frac{(n-2)180°}{n} = 180° - \frac{360°}{n}\]The polygon with more sides has the larger interior angle, so the difference is\[\left(180° - \frac{360°}{5x}\right) - \left(180° - \frac{360°}{4x}\right) = \frac{360°}{4x} - \frac{360°}{5x} = \frac{360°}{20x} = \frac{18°}{x}\]Setting this equal to \(9°\) gives \(x = 2\), so the polygons have 10 and 8 sides.Checking: a regular decagon has interior angle \(144°\) and a regular octagon \(135°\), a difference of \(9°\).

Q: What is the sum of the interior angles of a hexagon ?

Correct answer: 720 degreesThe interior angles of an \(n\)-sided polygon sum to \((n-2) \times 180^\circ\).\[(6-2) \times 180 = 4 \times 180 = 720^\circ\]For a regular hexagon each angle is therefore \(720 \div 6 = 120^\circ\).

Q: In a rhombus, the diagonals bisect each other at:

The diagonals of a rhombus bisect each other at right angles, cutting it into four congruent right triangles.\[ 90^{\circ} \]This is why the area can be written as \( \tfrac{1}{2}d_{1}d_{2} \).

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181.

The number of sides of two regular polygons are in the ratio 5:4 and the difference between their interior angles is 9°. Find the number of sides.

A.15, 12
B.20, 16
C.10, 8
D.5, 4

Answer: A

Correct answer: 10, 8

Let the polygons have \(5x\) and \(4x\) sides. Each interior angle of a regular \(n\)-gon is

\[\frac{(n-2)180°}{n} = 180° - \frac{360°}{n}\]

The polygon with more sides has the larger interior angle, so the difference is

\[\left(180° - \frac{360°}{5x}\right) - \left(180° - \frac{360°}{4x}\right) = \frac{360°}{4x} - \frac{360°}{5x} = \frac{360°}{20x} = \frac{18°}{x}\]

Setting this equal to \(9°\) gives \(x = 2\), so the polygons have 10 and 8 sides.

Checking: a regular decagon has interior angle \(144°\) and a regular octagon \(135°\), a difference of \(9°\).

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182.

What is the sum of the interior angles of a hexagon ?

A.540 degrees
B.720 degrees
C.900 degrees
D.1080 degrees

Answer: C

Correct answer: 720 degrees

The interior angles of an \(n\)-sided polygon sum to \((n-2) \times 180^\circ\).

\[(6-2) \times 180 = 4 \times 180 = 720^\circ\]

For a regular hexagon each angle is therefore \(720 \div 6 = 120^\circ\).

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183.

The point where the perpendicular bisectors of the sides of a triangle intersect is known as the:

A.Centroid
B.Incenter
C.Circumcenter
D.Orthocenter

Answer: C

The intersection point of the perpendicular bisectors of the sides of a triangle is the circumcenter. This point is equidistant from all three vertices of the triangle and is the center of the circle that circumscribes the triangle.

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184.

The point of concurrency of the angle bisectors of a triangle is called the:

A.Centroid
B.Incenter
C.Circumcenter
D.Orthocenter

Answer: B

The intersection point of the angle bisectors of the angles of a triangle is the incenter. This point is equidistant from all three sides of the triangle and is the center of the circle that is inscribed within the triangle.

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185.

In a rhombus, the diagonals bisect each other at:

A.30°
B.45°
C.60°
D.90°

Answer: D

The diagonals of a rhombus bisect each other at right angles, cutting it into four congruent right triangles.

\[ 90^{\circ} \]

This is why the area can be written as \( \tfrac{1}{2}d_{1}d_{2} \).

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186.

A triangle with vertices at (0,0), (4,0), and (2,3) is a/an:

A.Equilateral triangle
B.Isosceles triangle
C.Scalene triangle
D.Right-angled triangle

Answer: B

We use the distance formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to find the lengths of the sides.

Side 1 (between (0,0) and (4,0)): \(\sqrt{(4-0)^2} = 4\).

Side 2 (between (0,0) and (2,3)): \(\sqrt{(2-0)^2 + (3-0)^2} = \sqrt{4+9} = \sqrt{13}\).

Side 3 (between (4,0) and (2,3)): \(\sqrt{(2-4)^2 + (3-0)^2} = \sqrt{(-2)^2+3^2} = \sqrt{4+9} = \sqrt{13}\).

Since two sides have the same length (\(\sqrt{13}\)), the triangle is isosceles.

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187.

A line segment has no definite:

A.Length
B.Breadth
C.Endpoints
D.Direction

Answer: B

A line segment is one-dimensional: it has length and two endpoints, but no breadth and no thickness.

Length and endpoints are precisely what it does have, which is what makes breadth the odd one out.

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188.

In \(\triangle PQR\) and \(\triangle XYZ\), if \(PQ/XY = QR/YZ = PR/XZ\), which similarity criterion applies?

A.Aaa (angle-angle-angle)
B.Sas (side-angle-side)
C.Sss (side-side-side)
D.Rhs (right-hypotenuse-side)

Answer: C

The condition given is that the ratios of the lengths of the corresponding sides of the two triangles are equal. This is the definition of the SSS (Side-Side-Side) similarity criterion. If this condition is met, the two triangles are similar.

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Geometry (Lines, Angles, Triangles, Polygons) Questions and Answers

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Geometry (Lines, Angles, Triangles, Polygons) Questions and Answers

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Tags: Geometry Lines, Angles, Triangles Questions || Polygons MCQ Questions and Answers || Geometry Quantitative Aptitude || Angles and Triangles GK Questions || Quantitative Aptitude Geometry GK

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