Geometry and Mensuration Questions for CAT

In the given figure $AB$ is the diameter of the circle and $\angle PAB = 25^{\circ}$. Find $\angle TPA$.

Three identical circles touch each other externally. If the tangents at their points of contact meet at a point whose distance from any point of contact is 4 cm, then what is the radius (in cm) of circles?

Let $L_1(n)$ be the perimeter of the regular $n$‑gon circumscribing a circle of radius 1 cm, and $L_2(n)$ be the perimeter of the regular $n$‑gon inscribed in the same circle. Then what is the value of $\frac{L_1(13) + 2\pi}{L_2(17)}$?

A man walks 9 km east, 12 km north, then 5 km west. Find net displacement.

$ABCD$ has area equal to $28$. $BC$ is parallel to $AD$. $BA$ is perpendicular to $AD$. If $BC = 6$ and $AD = 8$, then what is $AB$?

In the given diagram, if $O$ is the center of the circle, chord $PR$ and $SQ$ intersect each other at $O$, $\angle POQ = 80^\circ$, then $\angle QSR =\ ?$

A ladder reaches a window that is 8 m above the ground on one side of the street. Keeping its foot on the same point, the ladder is twined to the other side of the street to reach a window 12 m high. Find the width of the street if the ladder is 13 m.

Two circles with radii 4 cm and 12 cm, centres 28 cm apart. Find length of common transverse tangent PQ.

In a triangle $ABC$, point $D$ is on side $AB$ and point $E$ is on side $AC$, such that $BCED$ is a trapezium. $DE : BC = 3 : 5$. Calculate the ratio of the area of $\triangle ADE$ and the trapezium $BCED$.

A square sheet of side 10 m is taken and 4 squares of side 2 m each are cut from the 4 corners of the sheet. The remaining part is folded to form a cuboid with one open end. Find the volume of the cuboid.

In trapezium DEFG, DE ∥ FG. If DE = 2.4 cm, FG = 1.2 cm, and the perimeter of trapezium is 8.4 cm, then the perimeter (in cm) of △DEP, where DG and EF extended meet at P, is:

Quarter circles are drawn inside a square of side 8 cm with vertices as centers and radius equal to 8 cm. A circle is drawn inside the square tangential to all the four quarter circles. Find the radius of the circle.

A rectangle of maximum area is inscribed in a semicircle of diameter $10$ cm. Find the ratio of its length to breadth.

A rectangular slate measuring $5 m \times 20 cm$ floats. It sinks $2 cm$ when a stone is added. The density of water is $1000 kg/m^3$. What is the mass of the stone?

A. $40 kg$

B. $20 kg$

C. $2 kg$

D. $200 kg$

A ping pong ball is dropped from a 45 metres high building. The ball bounces back three-fifths of the distance each time before coming to rest. What is the total distance traversed by the ball?

A rectangle PQRS has area 960 sq cm. From it, a semicircular region with diameter PQ and area 50π sq cm is cut out. Find the perimeter of the remaining figure.

$l_1 \parallel l_2$. A path goes from $A$ on $l_1$ to a point $B$ between the lines, then to $C$ on $l_2$. The interior angles that the path makes at $A$ (with $l_1$) and at $C$ (with $l_2$), both measured on the side facing $B$, are $132°$ and $116°$. Find $\angle ABC$ (in degrees).

In triangle $ABC$, medians $AD = 9$ cm and $BE = 16$ cm are perpendicular to each other. Find the area of triangle $ABC$ (in sq cm).

There is a triangular building (ABC) located in Jaipur. The wall BC $=1500$ ft. AB is a perfect cube, AC is a power of 3, and AC $=3\,\text{AB}$. Determine the perimeter of this triangular building.

Find the area of a triangle with sides 12, 17 and 25 (in square units).