Geometry

Two triangles are congruent if two angles and the included side of one are equal to two angles and the included side of the other. This is known as:

If two sides of a triangle are 7 cm and 10 cm, the third side must be less than:

In $\triangle ABC$, if $AB = BC = CA$, then each interior angle is:

The three sides of a triangle are 5 cm, 8 cm, and $x$ cm. Which of the following is a possible value for $x$?

Two sides of an isosceles triangle are 4 cm and 9 cm. What is the length of the third side?

In the given figure, $AC > AB$ and $AD$ is the bisector of $\angle BAC$. Which relationship between $\angle ADB$ and $\angle ADC$ is correct?

In $\triangle ABC$, $\angle B = 2\angle C$. $D$ is a point on $BC$ such that $AD$ bisects $\angle BAC$ and $AD = CD$. Find the measure of $\angle BAC$.

In the figure, $O$ is a point in the interior of $\triangle ABC$. Which of the following inequalities is always true regarding the sum of distances from $O$ to the vertices?

In $\triangle ABC$, the side $BC$ is produced to $D$. If the bisectors of $\angle ABC$ and $\angle ACD$ meet at $E$, and $\angle BAC = 70^\circ$, find $\angle BEC$.

In a triangle $ABC$, the medians $BE$ and $CF$ are equal. What type of triangle is $ABC$?

In the given figure, $D$ and $E$ are mid-points of $AB$ and $AC$. If $DE + BC = 18\text{ cm}$, find $BC$.

In $\triangle ABC$, $M$ is the mid-point of $AB$. A line through $M$ parallel to $BC$ meets $AC$ at $N$. If $AN = 3.5\text{ cm}$, find $AC$.

The diagonal $AC$ of a quadrilateral $ABCD$ is $16\text{ cm}$. If $P$ and $S$ are the mid-points of $AB$ and $AD$ respectively, find $PS$.

In $\triangle ABC$, the mid-points of $AB, BC$ and $CA$ are $D, E, F$. If $AB = 6, BC = 8, CA = 10$, find $EF$.

If the mid-points of sides of a square are joined in order, the resulting figure is a:

In $\triangle ABC$, $AD$ and $BE$ are medians. If $DF \parallel BE$ (where $F$ is on $AC$), find the ratio $EF : FC$.

$ABCD$ is a rectangle. $P, Q, R, S$ are mid-points of $AB, BC, CD, DA$. What type of quadrilateral is $PQRS$?

In $\triangle ABC$, $P$ and $Q$ are mid-points of $AB$ and $AC$. If the area of $\triangle APQ$ is $12$ cm$^2$, and $R$ is a point on $BC$ such that $BR = RC$, what is the area of the quadrilateral $PBRQ$?

A frame is shaped as a kite $ABCD$ where $AB = AD$ and $CB = CD$. $E, F, G,$ and $H$ are mid-points of sides $AB, BC, CD,$ and $DA$ respectively. Which of the following best describes the shape of the quadrilateral $EFGH$?

In a right-angled triangle $ABC$, right-angled at $B$, $D$ is the mid-point of $AC$. If a line $DE$ is drawn perpendicular to $BC$ meeting $BC$ at $E$, and $AB = 8$ cm, $BC = 6$ cm, find the length of $DE$.

In the given right-angled triangle $ABC$, the angle $\angle B = 90^\circ$. If the length of side $AB = 6$ cm and the length of side $BC = 8$ cm, find the length of the hypotenuse $AC$.

In $\triangle ABC$, $AB=AC=13$ cm and the altitude from $A$ to $BC$ is $12$ cm. Find $BC$.

A wire of length $25$ m is attached to a vertical pole of height $20$ m and is fixed to a stake on the ground. How far is the stake from the base of the pole?

In a square of side $10$ cm, find the length of the diagonal.

The sides of a triangle are $a, b, c$. If $a^2 + b^2 < c^2$, the triangle is:

In $\triangle ABC$, $D$ is the midpoint of $BC$. If $AB^2 + AC^2 = 2(AD^2 + BD^2)$, this is known as:

An airplane leaves an airport and flies due North at a speed of $1000$ km/hr. At the same time, another airplane leaves the same airport and flies due West at a speed of $1200$ km/hr. How far apart will the two planes be after $1.5$ hours?

A wire attached to a vertical pole of height $18$ m is $24$ m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?

In a triangle $ABC$, $AB^2 + AC^2 = BC^2$. If a line $AD$ is drawn such that $D$ is the midpoint of $BC$, which of the following is true regarding the length of $AD$?

1. Pythagoras Theorem and its Converse: practice question 1 for Geometry?

In the given parallelogram $ABCD$, if $\angle BCD = 115^∘$, what is the measure of $\angle DAB$?

If the angles of a quadrilateral are in the ratio $1:2:3:4$, find the largest angle.

In a square $ABCD$, if the side is $5$ cm, find the length of the diagonal $AC$.

The diagonals of which of the following quadrilaterals are always equal in length?

In a parallelogram $ABCD$, $AB = 10$ cm and $AD = 6$ cm. The bisector of $\angle A$ meets $DC$ at $E$. Find the length of $DE$.

In a rectangle $ABCD$, diagonal $AC$ is inclined to $AB$ at $30^∘$. Find the acute angle between the diagonals.

In a parallelogram $ABCD$, if $\angle A = (2x + 10)^∘$ and $\angle C = (3x - 20)^∘$, find the value of $x$.

In a rectangle $ABCD$, point $P$ lies on side $BC$ such that $BP : PC = 1 : 2$. If the area of $\triangle ABP$ is $15\text{ cm}^2$, find the area of the rectangle $ABCD$.

A floor is paved with $200$ tiles, each in the shape of a rhombus. The diagonals of each tile are $40\text{ cm}$ and $25\text{ cm}$. Find the total cost of polishing the floor at Rs $4$ per square meter.

In a parallelogram $ABCD$, $E$ is the midpoint of $AB$ and $F$ is the midpoint of $CD$. If the area of the parallelogram $ABCD$ is $48\text{ cm}^2$, find the area of the quadrilateral $AECF$.

In the figure, triangles $ABC$ and $ABD$ are on the same base $AB$ and between the same parallels $AB$ and $CD$. If the area of $\triangle ABC$ is $35 \text{ cm}^2$, what is the area of $\triangle ABD$?

In the given figure, $ABCD$ is a parallelogram and $\triangle ABE$ is a triangle on the same base $AB$ and between the same parallels $AB$ and $CD$. If the area of parallelogram $ABCD$ is $48 \text{ cm}^2$, find the area of $\triangle ABE$.

In $\triangle ABC$, $D, E, F$ are the midpoints of sides $BC, CA, AB$ respectively. Then $ar(DEF)$ is:

Two parallelograms are on the same base and between the same parallels. The ratio of their areas is:

If the area of a parallelogram $ABCD$ is $100\text{ cm}^2$, and $L, M$ are midpoints of $AB$ and $CD$, find $ar(LMCB)$.

In the given figure, $PQRS$ is a square and $T$ and $U$ are the midpoints of $PS$ and $QR$ respectively. Find the ratio $ar(\triangle PQT) : ar(PQRS)$.

A rectangle and a triangle have the same base and same area. If the height of the rectangle is $10\text{ cm}$, what is the height of the triangle?

A landscape designer is planning a park within a parallelogram $ABCD$. A point $P$ is chosen inside the parallelogram such that the area of $\triangle APB$ is $45$ $m^2$ and the area of $\triangle CPD$ is $25$ $m^2$. If the distance between the parallel lines $AD$ and $BC$ is $10$ $m$, calculate the length of the side $AD$.

Two triangles $ABC$ and $DBC$ are on the same base $BC$ and on the same side of $BC$ such that $AD \parallel BC$. If the area of $\triangle ABC$ is $35 \text{ cm}^2$, what is the area of $\triangle DBC$?

In the given figure, $ABCD$ is a parallelogram and $P$ is a point inside it. Prove that the sum of the areas of $\triangle PAB$ and $\triangle PCD$ is equal to:

In a circle, two chords $PQ$ and $RS$ are equidistant from the center. If $PQ = 10$ cm, find the length of $RS$.

An equilateral triangle $ABC$ is inscribed in a circle with center $O$. Find $\angle BOC$.

What is the angle subtended by a semi-circle at any point on the remaining part of the circle?

If the length of a chord is $24$ cm and the radius is $13$ cm, find the distance of the chord from the center.

Two chords $AB$ and $CD$ of lengths $5$ cm and $11$ cm respectively are parallel and are on opposite sides of the center $O$. If the distance between $AB$ and $CD$ is $6$ cm, find the radius of the circle.

If a line intersects two concentric circles with center $O$ at $A, B, C$ and $D$, and $AD = 12$ cm, $BC = 8$ cm, find $AB$.

In the figure, $O$ is the center of the circle and $AD$ is a diameter. If $AB = BC = CD$, then $\angle AOB$ is:

Points $P, Q, R$ are on a circle such that $\angle PQR = 100^\circ$. If $O$ is the center of the circle, find the measure of $\angle OPR$.

In a circle, chords $AB$ and $CD$ are equal and intersect at point $E$. If $AB = 10$ cm and the distance of $AB$ from center $O$ is $6$ cm, find the distance of chord $CD$ from the center $O$.

A chord $MN$ of a circle of radius $15$ cm subtends a right angle at the center $O$. Calculate the length of the chord $MN$.