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Geometry and Mensuration PYQs for GATE DA

Solve 1+ Geometry and Mensuration previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

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Question 1

2026 PYQ

Level 3: Exam Standard

In the given figure, P, Q, and R are three points on a circle of radius 10 cm with O as its center, \overline{P Q} = \overline{R Q}, and ∠ P QR = 4 5^{\circ} . The figure is representative. 45° Q P R O The area of the shaded region P QR O is ______________c m^{2} .

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Geometry and Mensuration PYQs for GATE DA

Solve 1+ Geometry and Mensuration previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Geometry and Mensuration

Your journey through spatial reasoning and measurement. Master these to decode any geometric configuration efficiently.

1. Circle Geometry and Chord-Angle Relations

Current Topic. Central angles, arcs, and segment properties.

2. Tangents, Secants, and Circle Theorems

Upcoming. Tangent properties, alternate segment theorem, and intersecting chords.

3. Area and Perimeter of 2D Figures

Upcoming. Triangles, quadrilaterals, and composite shapes.

4. Surface Area and Volume of 3D Solids

Upcoming. Prisms, cylinders, cones, and spheres with unit conversions.

What You Will Master

Ability to quickly decode geometric configurations, apply chord and tangent theorems, and compute measurements without getting trapped by similar-figure ratios or hidden isosceles triangles.

Circle Geometry and Chord-Angle Relations

A circle is the set of all points in a plane that are at a fixed distance (the radius) from a central point (the center).

When we connect points on the circumference, we create:

Chord: A line segment connecting any two points on the circle.
Arc: A continuous portion of the circumference.
Segment: The region bounded by a chord and an arc.

The core of circle geometry lies in understanding how the angles formed by these elements relate to one another, specifically the relationship between the angle at the center and the angle at the circumference.

Geometry and Mensuration: Solved Questions with Step-by-Step Explanations (1 Problems)

Question 1 · Quantitative Aptitude · 2026 MCQ

In the given figure,

P, Q,

and

R

are three points on a circle of radius 10 cm with

O

as its center,

\overline{P Q} = \overline{R Q}

, and

∠ P QR = 4 5^{\circ}

. The figure is representative.

The area of the shaded region

P QR O

is ______________

c m^{2}

A.
50
B.
$252$
C.
$502$
D.
100

Correct Answer:

Step-by-Step Solution

Insight: The quadrilateral PQRO is a kite with perpendicular diagonals, making its area simply half the product of the diagonals.

Exam route:

By the Central Angle Theorem, $∠ P O R = 2 \times ∠ P QR = 2 \times 4 5^{\circ} = 9 0^{\circ}$ .
$△ P O R$ is a right isosceles triangle with legs $O P = O R = 10$ . Thus, hypotenuse $P R = 1 0^{2} + 1 0^{2} = 102$ .
Since $P Q = R Q$ and $O P = O R$ , the line $O Q$ is the perpendicular bisector of $P R$ .
The area of quadrilateral $P QR O$ is $\frac{1}{2} \times diagonal_{1} \times diagonal_{2} = \frac{1}{2} \times P R \times O Q$ .
$O Q$ is a radius, so $O Q = 10$ .
Area = $\frac{1}{2} \times 102 \times 10 = 502$ .

Learning route:

Identify the given: Circle with center $O$ , radius $r = 10$ . Points $P, Q, R$ on circle. $P Q = R Q$ , $∠ P QR = 4 5^{\circ}$ .
By the Central Angle Theorem, the angle subtended by arc $P R$ at the center is $∠ P O R = 2 \times ∠ P QR = 9 0^{\circ}$ .
In $△ P O R$ , $O P = O R = 10$ (radii) and $∠ P O R = 9 0^{\circ}$ . By Pythagoras, $P R = 1 0^{2} + 1 0^{2} = 102$ .
The quadrilateral $P QR O$ is a kite because adjacent sides are equal ( $O P = O R = 10$ , and $P Q = R Q$ is given).
A key property of a kite is that its diagonals are perpendicular. Thus, $O Q ⊥ P R$ .
The area of any quadrilateral with perpendicular diagonals is $\frac{1}{2} \times d_{1} \times d_{2}$ .
Here, $d_{1} = P R = 102$ , and $d_{2} = O Q = 10$ (since $Q$ is on the circle, $O Q$ is a radius).
Area = $\frac{1}{2} \times 102 \times 10 = 502 cm^{2}$ .

Try a question

Login to view the complete previous-year questions and solutions

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

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Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Geometry and Mensuration PYQs for GATE DA

Chapter Roadmap: Geometry and Mensuration

Chapter Roadmap: Geometry and Mensuration

Circle Geometry and Chord-Angle Relations

Circle Geometry and Chord-Angle Relations

Geometry and Mensuration: Solved Questions with Step-by-Step Explanations (1 Problems)

Step-by-Step Solution

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