Triangles: 2 marks Questions (CBSE Class 10)

In the given figure, QR $\parallel$ CB and RP $\parallel$ AC. If BR = 10 cm, QA = 12 cm, BP = 12 cm and PC = 18 cm, then find the lengths of AR and QC.

In the given figure, AB $\parallel$ DC. If OB = 3OD and CD = 1.8 cm, then find the length AB.

In the given figure, $△ODC\sim △OBA$. If $\angle BOC=110^{\circ }$, $\angle ODC=45^{\circ }$ and AB = 2 CD, then find (i) $m\angle OAB$ (ii) OB : OD.

In the given figure, two triangles ABC and PQR are shown such that $\angle A=\angle P$ and $\angle C=\angle R$. If AD $\perp$ BC and PS $\perp$ QR, then prove that (i) $△ADB\sim △PSQ$ (ii) AD $\times$ QS = BD $\times$ PS.

In $△ABC$, $DE\parallel BC$. If $AD=x$, $DB=x-2$, $AE=x+2$ and $EC=x-1$, then find the value of $x$.

In the figure given above, $△ABC\sim △XYZ$, then find the values of $x$ and $y$.

In the given figure, $△AHK\sim △ABC$. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find the length of AC.

In the given figure, XY $\parallel$ QR, $\frac{PQ}{XQ}=\frac{7}{3}$ and PR = 6.3 cm. Find the length of YR.

The diagonals of a quadrilateral ABCD intersect each other at the point O such that $\frac{AO}{OC}=\frac{BO}{OD}$. Show that quadrilateral ABCD is a trapezium.

Two right triangles PRQ and PSQ are drawn on the same hypotenuse PQ. If PR and QS intersect at T, prove that ST $\times$ TQ = PT $\times$ TR.

In the given figure, $\Delta ABE\cong \Delta ACD$. Prove that $\Delta ADE\sim \Delta ABC$.

D is a point on the side BC of $△ABC$ such that $\angle CAB=\angle CDA$. Show that $C{A}^2=CB\times CD$.

In the given figure, DEFG is a square. $△ABC$ is right angle triangle with $\angle A=90^{\circ }$. Prove that $AG\times DG=AF\times DB$.

In the given figure, $AB\parallel DE$ and $AC\parallel DF$. Show that $△ABC\sim △DEF$. If BC = 10 cm, EB = CF = 5 cm and AB = 7 cm, then find the length DE.

In the given figure, if PQ $\parallel$ RS, then prove that $△POQ\sim △SOR$.

In the given figure, $△OSR\sim △OQP$, $\angle ROQ=125^{\circ }$ and $\angle ORS=70^{\circ }$. Find the measures of $\angle OSR$ and $\angle OQP$.

D is a point on side BC of $△ABC$ such that $\angle ADC=\angle BAC$. Prove that $(CA{)}^2=CB\cdot CD$.

A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a shadow of 45 m on the ground. Find the height of the tower.

In the given figure $\angle ADE=\angle ACB$ and $\frac{AD}{DB}=\frac{AE}{EC}$. Prove that $△ABC$ is an isosceles triangle.

Two chords BA and CD intersect at point P outside the circle. Prove that $△PDA\sim △PBC$

The diagonal BD of parallelogram ABCD is divided by segment AE in the ratio 1 : 2. If BE = 1.8 cm, find the length of AD.

In a trapezium ABCD, AB $\parallel$ DC and its diagonals intersect at O. Prove that $\frac{OA}{OC}=\frac{OB}{OD}$.

In the given figure, AB $\parallel$ DE and BD $\parallel$ EF. Prove that DC${}^2$ = CF $\times$ AC.

The perimeters of two similar triangles are 22 cm and 33 cm respectively. If one side of first triangle is 9 cm, then find the length of corresponding side of the second triangle.

If $△ABC\sim △PQR$ in which $AB=6$ cm, $BC=4$ cm, $AC=8$ cm and $PR=6$ cm, then find the length of $(PQ+QR)$.

In the given figure, $\frac{QR}{QS}=\frac{QT}{PR}$ and $\angle 1=\angle 2$, show that $△PQS\sim △TQR$.

In the given figure, D is a point on the side BC of $△ABC$ such that $\angle ADC=\angle BAC$. Show that $C{A}^2=CD\cdot CB$.

In the given figure, $OA\cdot OB=OC\cdot OD$. Show that $\angle A=\angle C$ and $\angle B=\angle D$.

In the given figure, $\frac{PS}{SQ}=\frac{PT}{TR}$ and $\angle PST=\angle PRQ$. Prove that $△PQR$ is an isosceles triangle.

In the given figure, $△ABE\cong △ACD$. Prove that $△ADE\sim △ABC$.

A 1.5 m tall boy is walking away from the base of a lamp post which is 12 m high, at the speed of 2.5 m/sec. Find the length of his shadow after 3 seconds.

In parallelogram ABCD, side AD is produced to a point E and BE intersects CD at F. Prove that $△ABE\sim △CFB$

AD and PS are medians of triangles ABC and PQR respectively such that $△ABD\sim △PQS$. Prove that $△ABC\sim △PQR$.

AD and PS are angle bisectors of $\angle A$ and $\angle P$ of triangles ABC and PQR. If $△ABC\sim △PQR$, prove that $△ACD\sim △PRS$.

In $△ABC$ and $△PQR$, AD and PS are altitudes such that $△ABD\sim △PQS$ and $△ACD\sim △PRS$. Prove that $△ABC\sim △PQR$.

P is a point on the side BC of $△ABC$ such that $\angle APC=\angle BAC$. Prove that $A{C}^2=BC\cdot CP$.

In the adjoining figure, $\frac{AD}{BD}=\frac{AE}{EC}$ and $\angle BDE=\angle CED$, prove that $△ABC$ is an isosceles triangle.

In the adjoining figure, $AP=1$ cm, $BP=2$ cm, $AQ=1.5$ cm and $AC=4.5$ cm.
Prove that $△APQ\sim △ABC$. Hence find the length of PQ, if $BC=3.6$ cm.

In the given figure, $AP\perp AB$ and $BQ\perp AB$. If OA = 15 cm, BO = 12 cm and AP = 10 cm, then find the length of BQ.

D is a point on the side BC of $△ABC$ such that $\angle ADC=\angle BAC$. Show that $A{C}^2=BC\times DC$.

In the given figure, $OA\cdot OB=OC\cdot OD$. Prove that $△AOD\sim △COB$.

In the given figure, $\angle D=\angle E$ and $\frac{AD}{DB}=\frac{AE}{EC}$. Prove that $△ABC$ is isosceles.

In the given figure, $PQ\parallel RS$. Prove that $OP\times OR=OQ\times OS$.

In the given figure, ABCD is a quadrilateral. Diagonal BD bisects $\angle B$ and $\angle D$ both.
Prove that :
(i) $△ABD\sim △CBD$
(ii) AB = BC

Diagonals AC and BD of a trapezium ABCD intersect at O, where AB $\parallel$ DC. If $\frac{DO}{OB}=\frac{1}{2}$, then show that AB = 2CD

In $△ABC$, altitudes AD and BE are drawn. If AD = 7 cm, BE = 9 cm and EC = 12 cm then, find the length of CD.

In the given figure, $\frac{EA}{EC}=\frac{EB}{ED}$, prove that $△EAB\sim △ECD$

In the given figure, $△AHK\sim △ABC$. If AK = 8 cm, BC = 3.2 cm and HK = 6.4 cm, then find the length of AC.

In the given figure, ABC and AMP are two right triangles, right angled at B and M, respectively. Prove that $△ABC\sim △AMP$.

In the given figure, $DE\parallel AC$ and $\frac{BE}{EC}=\frac{BC}{CP}$. Prove that $DC\parallel AP$.

In the given figure, $LM\parallel CB$ and $LN\parallel CD$. Prove that $\frac{AM}{AN}=\frac{AB}{AD}$.

In the adjoining figure, A, B and C are points on OP, OQ and OR respectively such that $AB\parallel PQ$ and $AC\parallel PR$. Show that $BC\parallel QR$.

In the given figure, XZ is parallel to BC. AZ = 3 cm, ZC = 2 cm, BM = 3 cm and MC = 5 cm. Find the length of XY.

In the given figure, ABCD is a parallelogram. AE divides the line segment BD in the ratio 1 : 2. If BE = 1.5 cm, then find the length of BC.

In the given figure, ABC is a triangle in which DE$\parallel$BC. If AD = $x$, DB = $x-2$, AE = $x+2$ and EC = $x-1$, then find the value of $x$.

Diagonals AC and BD of trapezium ABCD with AB$\parallel$DC intersect each other at point O. Show that $\frac{OA}{OC}=\frac{OB}{OD}$.