NCERT Solutions
Class 10 - Mathematics

Chapter 6: TRIANGLES

Fill in the blanks using the correct word given in brackets:

1. All circles are ________. (congruent, similar)
2. All squares are ________. (similar, congruent)
3. All ________ triangles are similar. (isosceles, equilateral)
4. Two polygons of the same number of sides are similar, if (a) their corresponding angles are ________ and (b) their corresponding sides are ________. (equal, proportional)

Give two different examples of pair of:

(i) similar figures.

(ii) non-similar figures.

State whether the following quadrilaterals are similar or not (see Fig. 6.8).

In Fig. 6.17, (i) and (ii), DE \(\parallel\) BC. Find EC in (i) and AD in (ii).

E and F are points on the sides PQ and PR respectively of a \(\triangle PQR\). For each of the following cases, state whether \(EF \parallel QR\):

1. PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
2. PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
3. PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

In Fig. 6.18, if LM \(\parallel\) CB and LN \(\parallel\) CD, prove that

\[\dfrac{AM}{AB} = \dfrac{AN}{AD}.\]

In Fig. 6.19, DE \(\parallel\) AC and DF \(\parallel\) AE. Prove that

\[\dfrac{BF}{FE} = \dfrac{BE}{EC}.\]

In Fig. 6.20, DE \(\parallel\) OQ and DF \(\parallel\) OR. Show that \(EF \parallel QR\).

In Fig. 6.21, 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.

Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

ABCD is a trapezium in which AB \(\parallel\) DC and its diagonals intersect each other at the point O. Show that

\[\dfrac{AO}{BO} = \dfrac{CO}{DO}.\]

The diagonals of a quadrilateral ABCD intersect each other at the point O such that

\[\dfrac{AO}{BO} = \dfrac{CO}{DO}.\]

Show that ABCD is a trapezium.

State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in symbolic form:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

In Fig. 6.35, \(\triangle ODC \sim \triangle OBA\), \(\angle BOC = 125^\circ\) and \(\angle CDO = 70^\circ\). Find \(\angle DOC\), \(\angle DCO\) and \(\angle OAB\).

Diagonals AC and BD of a trapezium ABCD with \(AB \parallel DC\) intersect each other at the point O. Using a similarity criterion for two triangles, show that

\[ \dfrac{OA}{OC} = \dfrac{OB}{OD}. \]

In Fig. 6.36, \(\dfrac{QR}{QS} = \dfrac{QT}{PR}\) and \(\angle 1 = \angle 2\). Show that \(\triangle PQS \sim \triangle TQR\).

S and T are points on sides PR and QR of \(\triangle PQR\) such that \(\angle P = \angle RTS\). Show that \(\triangle RPQ \sim \triangle RTS\).

In Fig. 6.37, if \(\triangle ABE \cong \triangle ACD\), show that \(\triangle ADE \sim \triangle ABC\).

In Fig. 6.38, altitudes AD and CE of \(\triangle ABC\) intersect at P. Show that:

(i) \(\triangle AEP \sim \triangle CDP\)

(ii) \(\triangle ABD \sim \triangle CBE\)

(iii) \(\triangle AEP \sim \triangle ADB\)

(iv) \(\triangle PDC \sim \triangle BEC\)

E is a point on the side AD (produced) of a parallelogram ABCD and BE intersects CD at F. Show that \(\triangle ABE \sim \triangle CFB\).

In Fig. 6.39, ABC and AMP are two right triangles right-angled at B and M respectively. Prove that:

(i) \(\triangle ABC \sim \triangle AMP\)

(ii) \(\dfrac{CA}{PA} = \dfrac{BC}{MP}\)

CD and GH are respectively the bisectors of \(\angle ACB\) and \(\angle EGF\) such that D and H lie on sides AB and FE of \(\triangle ABC\) and \(\triangle EFG\) respectively. If \(\triangle ABC \sim \triangle FEG\), show that:

(i) \(\dfrac{CD}{GH} = \dfrac{AC}{FG}\)

(ii) \(\triangle DCB \sim \triangle HGE\)

(iii) \(\triangle DCA \sim \triangle HGF\)

In Fig. 6.40, E is a point on side CB of an isosceles triangle ABC with \(AB = AC\). If AD ⟂ BC and EF ⟂ AC, prove that \(\triangle ABD \sim \triangle ECF\).

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of \(\triangle PQR\) (see Fig. 6.41). Show that \(\triangle ABC \sim \triangle PQR\).

D is a point on side BC of a triangle ABC such that \(\angle ADC = \angle BAC\). Show that \(CA^2 = CB \cdot CD\).

Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR. Show that \(\triangle ABC \sim \triangle PQR\).

A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

If AD and PM are medians of triangles ABC and PQR, respectively, where \(\triangle ABC \sim \triangle PQR\), prove that

\[ \dfrac{AB}{PQ} = \dfrac{AD}{PM}. \]