Geometry Aptitude Questions & Formulas with answer

Ques 1 GATE 2025 SET-2

Three villages P, Q, and R are located in such a way that the distance PQ=13 km, QR=14 km and RP=15 km as shown in the figure. A straight road joins Q and R. It is proposed to connect P to this road QR by constructing another road. What is the minimum possible length (in km) of this connecting road?
Note: The figure shown is representative.

Ques 2 GATE 2025 SET-2

In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.

Ques 3 GATE 2025 SET-1

A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?

Ques 4 GATE 2025 SET-1

A circle with center at (x,y)=(0.5,0) and radius =0.5 intersects with another circle with center at (x,y)=(1,1) and radius =1 at two points. One of the points of intersection (x, y) is:

Ques 5 GATE 2025 SET-1

An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of 2π/n, is identical to the original.
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen?
Note: The figures shown are representative.

A

B

C

D

Ques 6 GATE 2025 SET-1

In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative.

Ques 7 GATE 2025

A rectangle has a length L and a width W, where L>W. If the width, W, is increased by 10%, which one of the following statements is correct for all values of L and W?

Ques 8 GATE 2025

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius r cm as shown in the figure. The side of the dodecagon is d cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side r cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.

Ques 9 GATE 2025

In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative.

Ques 10 GATE 2025

A stick of length one meter is broken at two locations at distances of b₁ and b₂ from the origin (0), as shown in the figure. Note that 012<1. Which one of the following is NOT a necessary condition for forming a triangle using the three pieces?
Note: All lengths are in meter. The figure shown is representative.

Ques 11 GATE 2025

A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?

Ques 12 GATE 2025

A circle with center at (x,y)=(0.5,0) and radius =0.5 intersects with another circle with center at (x,y)=(1,1) and radius =1 at two points. One of the points of intersection (x,y) is:

Ques 13 GATE 2025

An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of 2π/n, is identical to the original.
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen?
Note: The figures shown are representative.

A

B

C

D