NCERT Solutions for Class 10 Math Chapter 6 - Triangles

Question 1:

All circles are ________. (congruent, similar)

Question 2:

All squares are ________. (similar, congruent)

Question 3:

All ________ triangles are similar. (isosceles, equilateral)

Question 4:

Two polygons of the same number of sides are similar, if their corresponding angles are ________ and their corresponding sides are ________. (equal, proportional)

Question 5:

Give two different examples of a pair of: similar figures

Question 6:

Give two different examples of a pair of: non-similar figures.

Question 7:

State whether the following quadrilaterals are similar or not:

Question 8:

E and F are points on the sides PQ and PR respectively of a $∆$PQR. For each of the following cases, state whether EF || QR:
PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and
FR = 2.4 cm

Question 9:

E and F are points on the sides PQ and PR respectively of a $∆$PQR. For each of the following cases, state whether EF || QR: PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

Question 10:

E and F are points on the sides PQ and PR respectively of a $∆$PQR. For each of the following cases, state whether EF || QR: PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

Question 11:

Question 12:

In fig. DE || OQ and DF || OR. Show that EF || QR.

Question 13:

In fig., A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

Question 14:

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.

Question 15:

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.

Question 16:

Question 17:

Question 18:

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

Question 19:

In fig., $∆$ODC ~ $∆$OBA, $\angle$BOC = 125$°$ and
$\angle$CDO = 70º. Find $\angle$DOC, $\angle$DCO and
$\angle$OAB.

Question 20:

Question 21:

Question 22:

S and T are points on sides PR and QR of $∆$PQR such that $\angle$P = $\angle$RTS. Show that $∆$RPQ ~ $∆$RTS.

Question 23:

In figure, if $∆$ABE $\cong$ $∆$ACD, show that $∆$ADE ~ $∆$ABC

Question 24:

In fig., altitudes AD and CE of $∆$ABC intersect
each other at the point P. Show that:
(i) $∆$AEP ~ $∆$ CDP
(ii) $∆$ABD ~ $∆$CBE
(iii) $∆$AEP ~ $∆$ADB
(iv) $∆$PDC ~ $∆$BEC

Question 25:

E is a point on the side AD produced by a parallelogram ABCD and BE intersects CD at F. Show that $∆$ABE ~ $∆$CFB.

Question 26:

Question 27:

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 $∆$ABC and $∆$EFG respectively. If $∆$ABC ~ $∆$FEG, show that:

Question 28:

In fig., E is a point on side CB produced of an isosceles triangle ABC with AB = AC.
If AD $\perp$ BC and EF $\perp$ AC, prove that $∆$ABD ~ $∆$ECF.

Question 29:

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of $∆$PQR (see fig.). Show that $∆$ABC ~ $∆$PQR.

Question 30:

D is a point on the side BC of a triangle
ABC such that $\angle$ADC = $\angle$BAC. Show that
CA² = CB . CD.

Question 31:

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 $∆$ABC ~ $∆$PQR.

Question 32:

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.

Question 33:

Question 34:

Let $∆$ABC ~ $∆$DEF and their areas be respectively, 64 cm² and 121 cm². If EF = 15.4 cm, find BC.

Question 35:

Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.

Question 36:

In fig., ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that:

Question 37:

If the areas of two similar triangles are equal, prove that they are congruent.

Question 38:

D, E and F are respectively the mid-points of the sides AB, BC and CA of $∆$ABC. Find the ratio of the areas of $∆$DEF and $∆$ABC.

Question 39:

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

Question 40:

Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.

Question 41:

ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is

Question 42:

Sides of two similar triangles are in the ratio
4 : 9. Areas of these triangles are in the ratio

Question 43:

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
7 cm, 24 cm, 25 cm

Question 44:

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
3 cm, 8 cm, 6 cm

Question 45:

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
50 cm, 80 cm, 100 cm

Question 46:

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.
13 cm, 12 cm, 5 cm.

Question 47:

PQR is a triangle right angled at P and M is a point on QR such that PM $\perp$ QR. Show that PM² = QM.MR.

Question 48:

In fig., ABD is a triangle right angled at A and AC $\perp$ BD. Show that:

AB²= BC.BD

Question 49:

In fig., ABD is a triangle right angled at A and AC $\perp$ BD. Show that:

AC² = BC.DC

Question 50:

In fig., ABD is a triangle right angled at A and AC $\perp$ BD. Show that:

AD² = BD.CD

Question 51:

ABC is an isosceles triangle right angled at C.
Prove that AB² = 2AC².

Question 52:

ABC is an isosceles triangle with AC = BC.
If AB² = 2AC², prove that ABC is right triangle.

Question 53:

ABC is an equilateral triangle of side 2a. Find each of its altitudes.

Question 54:

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

Question 55:

In fig., O is a point in the interior of a triangle ABC, OD $\perp$ BC, OE $\perp$ AC and OF $\perp$ AB. Show that:

Question 56:

In fig., O is a point in the interior of a triangle ABC, OD $\perp$ BC, OE $\perp$ AC and OF $\perp$ AB. Show that:

Question 57:

A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.

Question 58:

A guy 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?

Question 59:

Question 60:

Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.

Question 61:

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C.
Prove that AE² + BD² = AB² + DE².

Question 62:

Question 63:

Question 64:

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

Question 65:

1. 120$°$
2. 60$°$
3. 90$°$
4. 45$°$

Question 66:

Question 67:

Question 68:

Question 69:

Question 70:

In fig., AD is a median of a triangle ABC and AM $\perp$ BC. Prove that:

Question 71:

In fig., AD is a median of a triangle ABC and AM $\perp$ BC. Prove that:

Question 72:

In fig., AD is a median of a triangle ABC and AM $\perp$ BC. Prove that:

Question 73:

Prove that sum of squares of the diagonals of a parallelogram is equal to sum of squares of its sides.

Question 74:

Question 75:

Question 76:

Question 77:

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out? If she pulls in the string at the rate of 5 cm per second, what will the horizontal distance of the fly from her after 12 seconds?