NCERT Exemplar Solutions
Class 10 - Mathematics

CHAPTER 6: Triangles

Is the triangle with sides 25 cm, 5 cm and 24 cm a right triangle? Give reasons.

It is given that \(\triangle DEF \sim \triangle RPQ\). Is it true that \(\angle D=\angle R\) and \(\angle F=\angle P\)? Why?

Points \(A\) and \(B\) lie on sides \(PQ\) and \(PR\) of \(\triangle PQR\) such that \(PQ=12.5\,\text{cm}\), \(PA=5\,\text{cm}\), \(PB=4\,\text{cm}\) and \(BR=6\,\text{cm}\). Is \(AB\parallel QR\)? Give reasons.

In Fig. 6.4, \(BD\) and \(CE\) intersect each other at \(P\). Is \(\triangle PBC \sim \triangle PDE\)? Why?

In triangles \(PQR\) and \(MST\), \(\angle P=55^\circ,\; \angle Q=25^\circ\) and \(\angle M=100^\circ,\; \angle S=25^\circ\). Is \(\triangle QPR \sim \triangle TSM\)? Why?

Is the following statement true? Why?

“Two quadrilaterals are similar if their corresponding angles are equal.”

Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of another triangle. Are the two triangles similar? Why?

If in two right triangles, one of the acute angles of one triangle is equal to an acute angle of the other triangle, can you say that the two triangles will be similar? Why?

The ratio of the corresponding altitudes of two similar triangles is \(\dfrac{3}{5}\). Is it correct to say that ratio of their areas is \(\dfrac{6}{5}\)? Why?

Point \(D\) is on side \(QR\) of \(\triangle PQR\) with \(PD\perp QR\). Is \(\triangle PQD \sim \triangle RPD\)? Why?

In Fig. 6.5, if \(\angle D = \angle C\), is it true that \(\triangle ADE \sim \triangle ACB\)? Why?

Is the following statement always true? “If an angle of one triangle equals an angle of another and two sides of one triangle are proportional to the corresponding two sides of the other triangle, then the triangles are similar.” Give reasons.

In \(\triangle PQR\), suppose \(PR^2 - PQ^2 = QR^2\) and \(M\) lies on \(PR\) with \(QM \perp PR\). Prove that \(QM^2 = PM \cdot MR\).

Find the value of \(x\) for which \(DE \parallel AB\) in Fig. 6.8.

In Fig. 6.9, if \(\angle 1=\angle 2\) and \(\triangle NSQ \cong \triangle MTR\), prove that \(\triangle PTS \sim \triangle PRQ\).

Diagonals of a trapezium \(PQRS\) intersect at \(O\). If \(PQ\parallel RS\) and \(PQ=3\,RS\), find \(\dfrac{\operatorname{ar}(\triangle POQ)}{\operatorname{ar}(\triangle ROS)}\).

In Fig. 6.10, if \(AB\parallel DC\) and lines \(AC\) and \(PQ\) meet at \(O\), prove that \(OA\cdot CQ = OC\cdot AP\).

Find the altitude of an equilateral triangle of side 8 cm.

If \(\triangle ABC \sim \triangle DEF\), with \(AB=4\,\text{cm}\), \(DE=6\,\text{cm}\), \(EF=9\,\text{cm}\) and \(FD=12\,\text{cm}\), find the perimeter of \(\triangle ABC\).

In Fig. 6.11, if \(DE\parallel BC\), find \(\operatorname{ar}(ADE):\operatorname{ar}(DECB)\).

In trapezium \(ABCD\) with \(AB\parallel DC\), points \(P\) and \(Q\) lie on \(AD\) and \(BC\) respectively, with \(PQ\parallel DC\). If \(PD=18\,\text{cm}\), \(BQ=35\,\text{cm}\) and \(QC=15\,\text{cm}\), find \(AD\).

Corresponding sides of two similar triangles are in the ratio \(2:3\). If the area of the smaller is \(48\,\text{cm}^2\), find the area of the larger triangle.

In \(\triangle PQR\), point \(N\) lies on \(PR\) with \(QN\perp PR\). If \(PN\cdot NR = QN^2\), prove that \(\angle PQR = 90^\circ\).

Areas of two similar triangles are \(36\,\text{cm}^2\) and \(100\,\text{cm}^2\). If a corresponding side of the larger is 20 cm, find the corresponding side of the smaller.

In Fig. 6.12, if \(\angle ACB = \angle CDA\), \(AC=8\,\text{cm}\) and \(AD=3\,\text{cm}\), find \(BD\).

A 15 m tower casts a 24 m shadow. At the same time a telephone pole casts a 16 m shadow. Find the pole’s height.

A 10 m ladder leans against a vertical wall with its foot 6 m from the wall. Find the height reached on the wall.

In Fig. 6.16, if \(\angle A = \angle C\), \(AB = 6\,\text{cm}\), \(BP = 15\,\text{cm}\), \(AP = 12\,\text{cm}\) and \(CP = 4\,\text{cm}\), find the lengths \(PD\) and \(CD\).

Given \(\triangle ABC \sim \triangle EDF\) with \(AB=5\,\text{cm}\), \(AC=7\,\text{cm}\), \(DE=12\,\text{cm}\), \(DF=15\,\text{cm}\). Find the remaining sides.

Prove: If a line is drawn parallel to one side of a triangle to meet the other two sides, then it divides those sides in the same ratio.

In Fig. 6.17, \(PQRS\) is a parallelogram and \(AB\parallel PS\). Prove that \(OC\parallel SR\).

A 5 m ladder reaches a wall at height 4 m. If the foot is moved \(1.6\,\text{m}\) towards the wall, by how much does the top slide up?

City route: \(AC\perp CB\), \(AC=2x\) km, \(CB=2(x+7)\) km. A direct highway \(AB=26\) km is planned. How many km are saved?

A flag pole 18 m high casts a 9.6 m shadow. Find the distance from the top of the pole to the far end of the shadow.

A lamp is on a 6 m pole. A 1.5 m woman casts a 3 m shadow. How far is she from the pole?

In Fig. 6.18, \(\triangle ABC\) is right-angled at \(B\) and \(BD\perp AC\). If \(AD=4\,\text{cm}\) and \(CD=5\,\text{cm}\), find \(BD\) and \(AB\).

In Fig. 6.19, \(\triangle PQR\) is right-angled at \(Q\) and \(QS\perp PR\). If \(PQ=6\,\text{cm}\), \(PS=4\,\text{cm}\), find \(QS\), \(RS\) and \(QR\).

In \(\triangle PQR\), let \(PD\perp QR\) with \(D\in QR\). If \(PQ=a\), \(PR=b\), \(QD=c\) and \(DR=d\), prove

\[(a+b)(a-b)=(c+d)(c-d).\]

In a quadrilateral \(ABCD\), \(\angle A+\angle D=90^\circ\). Prove that

\[AC^2+BD^2=AD^2+BC^2.\]

In Fig. 6.20, \(\ell\parallel m\) and the segments \(AB,\,CD,\,EF\) meet at \(P\). Prove

\[\dfrac{AE}{BF}=\dfrac{AC}{BD}=\dfrac{CE}{FD}.\]

In Fig. 6.21, \(PA, QB, RC, SD\) are all perpendicular to a line \(\ell\). Given \(AB=6\,\text{cm}\), \(BC=9\,\text{cm}\), \(CD=12\,\text{cm}\) and \(SP=36\,\text{cm}\). Find \(PQ, QR, RS\).

In a trapezium \(ABCD\) with \(AB\parallel DC\), diagonals \(AC\) and \(BD\) meet at \(O\). Through \(O\), draw \(PQ\parallel AB\) meeting \(AD\) at \(P\) and \(BC\) at \(Q\). Prove that \(PO=QO\).

In Fig. 6.22, segment \(DF\) meets \(AC\) at \(E\) in \(\triangle ABC\), where \(E\) is the midpoint of \(CA\) and \(\angle AEF=\angle AFE\). Prove

\[\dfrac{BD}{CD}=\dfrac{BF}{CE}.\]

Prove that the area of the semicircle on the hypotenuse of a right triangle equals the sum of the areas of the semicircles on the other two sides.

Prove that the area of the equilateral triangle on the hypotenuse of a right triangle equals the sum of the areas of the equilateral triangles on the other two sides.