CAT Geometry Previous Year Questions (PYQs)

Two tangents drawn from a point $P$ touch a circle with center $O$ at points $Q$ and $R$. Points $A$ and $B$ lie on $PQ$ and $PR$, respectively, such that $AB$ is also a tangent to the same circle. If $\angle AOB = 50^\circ$, then $\angle APB$, in degrees, equals

Let $ABCDEF$ be a regular hexagon and $P$ and $Q$ be the midpoints of $AB$ and $CD$, respectively. Then, the ratio of the areas of trapezium $PBCQ$ and hexagon $ABCDEF$ is

In a $\triangle ABC$, points $D$ and $E$ are on the sides $BC$ and $AC$, respectively. $BE$ and $AD$ intersect at point $T$ such that $AD:AT = 4:3$, and $BE:BT = 5:4$. Point $F$ lies on $AC$ such that $DF$ is parallel to $BE$. Then, $BD:CD$ is

In a circle with center $C$ and radius $6\sqrt{2}$ cm, $PQ$ and $SR$ are two parallel chords separated by one of the diameters. If $\angle PQC = 45^\circ$, and the ratio of the perpendicular distance of $PQ$ and $SR$ from $C$ is $3:2$, then the area, in sq. cm, of the quadrilateral $PQRS$ is

The $(x, y)$ coordinates of vertices P, Q and R of a parallelogram PQRS are $(-3, -2)$, $(1, -5)$ and $(9, 1)$, respectively. If the diagonal SQ intersects the $x$-axis at $(a, 0)$, then the value of $a$ is

If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is

In the XY-plane, the area, in sq. units, of the region defined by the inequalities $y \geq x+4$ and $-4 \leq x^{2}+y^{2}+4(x-y) \leq 0$ is

The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm, and the sum of the lengths of all its edges is 144 cm. The volume, in cubic cm, of the sphere is

ABCD is a rectangle with sides AB = 56 cm and BC = 45 cm, and E is the midpoint of side CD. Then, the length, in cm, of radius of in circle of $\triangle$ADE is

Three circles of equal radii touch (but not cross) each other externally. Two other circles, X and Y, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of X is more than that of Y, the ratio of the radii of X and Y is

The coordinates of the three vertices of a triangle are: $(1, 2)$, $(7, 2)$, and $(1, 10)$. Then the radius of the in circle of the triangle is

ABCD is a trapezium in which AB is parallel to CD. The sides AD and BC when extended, intersect at point E. If AB = 2 cm, CD = 1 cm, and perimeter of ABCD is 6 cm, then the perimeter, in cm, of $\triangle AEB$ is

A circular plot of land is divided into two regions by a chord of length $10\sqrt{3}$ meters such that the chord subtends an angle of $120^\circ$ at the center. Then, the area, in square meters, of the smaller region is

The midpoints of sides $AB$, $BC$, and $AC$ in $\triangle ABC$ are $M$, $N$, and $P$, respectively. The medians drawn from $A$, $B$, and $C$ intersect the line segments $MP$, $MN$ and $NP$ at $X$, $Y$, and $Z$, respectively. If the area of $\triangle ABC$ is $1440$ sq cm, then the area, in sq cm, of $\triangle XYZ$ is

A regular octagon ABCDEFGH has sides on length 6 cm each. Then the area, in sq. cm, of the square ACEG is

A quadrilateral $A B C D$ is inscribed in a circle such that $A B: C D=2: 1$ and $B C: A D=5: 4$. If $A C$ and $B D$ intersect at the point $E$, then $A E$ : $C E$ equals

Let $C$ be the circle $x^2 + y^2 + 4x - 6y - 3 = 0$ and $L$ be the locus of the point of intersection of a pair of tangents to $C$ with the angle between the two tangents equal to $60°$. Then, the point at which $L$ touches the line $x = 6$ is

In a right-angled triangle $\triangle A B C$, the altitude $A B$ is $5 \mathrm{~cm}$, and the base $B C$ is $12 \mathrm{~cm}$. $P$ and $Q$ are two points on $B C$ such that the areas of $\triangle \mathrm{ABP}, \triangle \mathrm{ABQ}$ and $\triangle \mathrm{ABC}$ are in arithmetic progression. If the area of $\triangle \mathrm{ABC}$ is $1.5$ times the area of $\triangle \mathrm{ABP}$, the length of PQ in cm , is

A triangle ABC is formed with AB = AC = 50 cm and BC = 80 cm. Then, the sum of the lengths, in cm, of all three altitudes of the triangle ABC is

ABCD is a trapezium in which AB is parallel to DC, AD is perpendicular to AB, and AB = 3DC. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area, in sq. cm, of the trapezium is

In $\triangle ABC$, $AB=AC=12$ cm and D is a point on side BC such that $AD=8$ cm. If AD is extended to a point E such that $\angle ACB=\angle AEB$, then the length, in cm, of AE is

A triangle is drawn with its vertices on the circle C such that one of its sides is a diameter of C and the other two sides have their lengths in the ratio $a: b$. If the radius of the circle is $r$, then the area of the triangle is

In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is

Let $\triangle ABC$ be an isosceles triangle such that $AB$ and $AC$ are of equal length. $AD$ is the altitude from $A$ on $BC$ and $BE$ is the altitude from $B$ on $AC$. If $AD$ and $BE$ intersect at $O$ such that $\angle AOB = 105^\circ$, then $AD/BE$ equals

A rectangle with the largest possible area is drawn inside a semicircle of radius $2$ cm. Then, the ratio of the lengths of the largest to the smallest side of this rectangle is