Top 80+ XAT Geometry Questions with Solutions PDF

An industrial robot manufacturing company is tasked to design humanoid robots to be used in warehouses where the robots need to pick items from a stack of shelves. The height of the topmost shelf from the ground is 7 feet. To operate, the robot has to move on a track, running parallel to the stack of shelves. The track is fixed 1 foot away from the base of the stack of shelves. Further, the robot cannot raise its arms by more than 60° from the horizontal plane.
If the robot’s arms are attached to its shoulder, what should be the minimum height of the robot from the ground to the shoulder for its arms to reach the topmost shelf?

A farmer has a quadrilateral parcel of land with a perimeter of 700 feet. Two opposite angles of that parcel of land are right angles, while the remaining two are not. The farmer wants to do organic farming on that parcel of land. The cost of organic farming is Rs. 400 per square foot.
Consider the following two additional pieces of information:
I. The length of one of the sides of that parcel of land is 110 feet.
II. The distance between the two corner points where the non-perpendicular sides of that parcel of land intersect is 255 feet.
To determine the amount of money the farmer needs to spend to do organic farming on the entire parcel of land, which of the above additional pieces of information is/are MINIMALLY SUFFICIENT?

ABCD is a rectangle, where the coordinates of C and D are (- 2,0) and (2,0), respectively.
If the area of the rectangle is 24, which of the following is a possible equation representing the line $$\overleftrightarrow{AB}$$?

Adu and Amu have bought two pieces of land on the Moon from an e-store. Both the pieces of land have the same perimeters, but Adu’s piece of land is in the shape of a square, while Amu’s piece of land is in the shape of a circle.
The ratio of the areas of Adu’s piece of land to Amu’s piece of land is:

A straight line $$L_1$$ has the equation $$y = k(x - 1)$$, where k is some real number. The straight line $$L_1$$ intersects another straight line $$L_2$$ at the point (5, 8).
If $$L_2$$ has a slope of 1, which of the following is definitely FALSE?

A soild trophy, consisting of two parts, has been designed in the following manner: the bottom part is a frustum of a cone with the bottom radius 30 cm, the top radius 20 cm, and height 40 cm, while the top part is a hemisphere with radius 20 cm. Moreover, the flat surface of the hemisphere is the same as the top surface of the frustum.
If the entire trophy is to be gold-plated at the cost of Rs. 40 per square cm, what would the cost for gold-plating be closest to?

Consider a right-angled triangle ABC, right angled at B. Two circles, each of radius r, are drawn inside the triangle in such a way that one of them touches AB and BC, while the other one touches AC and BC. The two circles also touch each other (see the image below).
If AB = 18 cm and BC = 24 cm, then find the value of r.

A farmer has a triangular plot of land. One side of the plot, henceforth called the base, is 300 feet long and the other two sides are equal. The perpendicular distance, from the corner of the plot, where the two equal sides meet, to the base, is 200 feet. To counter the adverse effect of climate change, the farmer wants to dig a circular pond. He plans that half of the circular area will be inside the triangular plot and the other half will be outside, which he will purchase at the market rate from his neighbour. The diameter of the circular plot is entirely contained in the base and the circumference of the pond touches the two equal sides of the triangle from inside.
If the market rate per square feet of land is Rs. 1400, how much does the farmer must pay to buy the land from his neighbour for the pond? (Choose the closest option.)

A group of boys is practising football in a rectangular ground. Raju and Ratan are standing at the two opposite mid-points of the two shorter sides. Raju has the ball, who passes it to Rivu, who is standing somewhere on one of the longer sides. Rivu holds the ball for 3 seconds and passes it to Ratan. Ratan holds the ball for 2 seconds and passes it back to Raju. The path of the ball from Raju to Rivu makes a right angle with the path of the ball from Rivu to Ratan. The speed of the ball, whenever passed, is always 10 metre per second, and the ball always moves on straight lines along the ground.
Consider the following two additional pieces of information:
I. The dimension of the ground is 80 metres × 50 metres.
II. The area of the triangle formed by Raju, Rivu and Ratan is 1000 square metres.
Consider the problem of computing the following: how many seconds does it take for Raju to get the ball back since he passed it to Rivu? Choose the correct option.

Find the value of
$$\frac{\sin^{6}15^{\circ} + \sin^{6}75^{\circ} + 6\sin^{2}15^{\circ}\sin^{2}75^{\circ}}{\sin^{4}15^{\circ} + \sin^{4}75^{\circ} + 5\sin^{2}15^{\circ}\sin^{2}75^{\circ}}$$

ABC is a triangle with BC=5. D is the foot of the perpendicular from A on BC. E is a point on CD such that BE=3. The value of $$AB^2 - AE^2 + 6CD$$ is:

ABC is a triangle and the coordinates of A, B and C are (a, b-2c), (a, b+4c) and (-2a,3c) respectively where a, b and c are positive numbers.
The area of the triangle ABC is:

A non-flying ant wants to travel from the bottom corner to the diagonally opposite top corner of a cubical room. The side of the room is 2 meters. What will be the minimum distance that the ant needs to travel?

ABCD is a trapezoid where BC is parallel to AD and perpendicular to AB. Kindly note that BC< AD. P is a point on AD such that CPD is an equilateral triangle. Q is a point on BC such that AQ is parallel to PC. If the area of the triangle CPD is $$4\sqrt{\ 3}$$, find the area of the triangle ABQ.

Ramesh and Reena are playing with triangle ABC. Ramesh draws a line that bisects $$\angle BAC$$; this line cuts BC at D. Reena then extends AD to a point P. In response, Ramesh joins B and P. Reena then announces that BD bisects $$\angle PBA$$, hat a surprise! Together, Ramesh and Reena find that BD= 6 cm, AC= 9 cm, DC= 5 cm, BP=8 cm, and DP = 5 cm.

How long is AP?

A tall tower has its base at point K. Three points A, B and C are located at distances of 4 metres, 8 metres and 16 metres respectively from K. The angles of elevation of the top of the tower from A and C are complementary.

What is the angle of elevation (in degrees) of the tower’s top from B?

The six faces of a wooden cube of side 6 cm are labelled A, B, C, D, E and F respectively. Three of these faces A, B, and C are each adjacent to the other two, and are painted red. The other three faces are not painted. Then, the wooden cube is neatly cut into 216 little cubes of equal size. How many of the little cubes have no sides painted?

ABC is a triangle with integer-valued sides AB = 1, BC >1, and CA >1. If D is the mid-point of AB, then, which of the following options is the closest to the maximum possible value of the angle ACD (in degrees)?

The topmost point of a perfectly vertical pole is marked A. The pole stands on a flat ground at point D. The points B and C are somewhere between A and D on the pole. From a point E, located on the ground at a certain distance from D, the points A, B and C are at angles of 60, 45 and 30 degrees respectively. What is AB : BC : CD?

Two circles P and Q, each of radius 2 cm, pass through each other’s centres. They intersect at points A and B. A circle R is drawn with diameter AB. What is the area of overlap (in square cm) between the circles R and P?