Solution of Chapter 6. Triangles (NCERT - Mathematics Book)

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

(i) All circles are __________. (congruent, similar)

(ii) All squares are ___________. (similar, congruent)

(iii) All ____________ triangles are similar. (isosceles, equilateral)

(iv) 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:

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

E and F are points on the sides PQ and PR respectively of a Δ PQR. For each of the following cases, state whether EF || QR :

(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

In Fig. 6.18, if LM || CB and LN || CD, prove that

In Fig. 6.19, DE || AC and DF || AE. Prove that

In Fig. 6.20, DE || OQ and DF || OR. Show thatEF || QR.

In Fig. 6.21, A, B and C are points on OP, OQ andOR respectively such that AB || PQ and AC || PR.Show that BC || QR.

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

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

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

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

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:

In Fig. 6.35, ΔODC ~ ΔOBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠ DOC, ∠ DCO and ∠ OAB

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

In Fig. 6.36, and ∠1 = ∠2. Show that Δ PQS ~ Δ TQR.

S and T are points on sides PR and QR of
Δ PQR such that ∠P = ∠RTS. Show that
Δ RPQ ~ Δ RTS

In Fig. 6.37, if Δ ABE ≅Δ ACD, show that
Δ ADE ~ Δ ABC.

In Fig. 6.38, altitudes AD and CE of Δ ABCintersect each other at the point P. Showthat:

(i) Δ AEP ~ Δ CDP

(ii) Δ ABD ~ Δ CBE

(iii) Δ AEP ~ Δ ADB

(iv) Δ PDC ~ Δ BEC

E is a point on the side AD produced of aparallelogram ABCD and BE intersects CD at F. Show that Δ ABE ~ Δ CFB

In Fig. 6.39, ABC and AMP are two righttriangles, right angled at B and Mrespectively. Prove that:

(i) Δ ABC ~ Δ AMP

(ii)

CD and GH are respectively the bisectorsof ∠ ACB and ∠ EGF in such a way that D and H lieon sides AB and FE of Δ ABC and Δ EFGrespectively. If Δ ABC ~ Δ FEG, show that:

(i)

(ii) Δ DCB ~ Δ HGE

(iii) Δ DCA ~ Δ HGF

In Fig. 6.40, E is a point on side CBproduced of an isosceles triangle ABCwith AB = AC. If AD ⊥ BC and EF ⊥ AC,prove that Δ ABD ~ Δ ECF

Sides AB and BC and median AD of atriangle ABC are respectively proportionalto sides PQ and QR and medianPM of
Δ PQR (see Fig. 6.41). Show that
Δ ABC ~ Δ PQR.

D is a point on the side BC of a triangleABC such that ∠ ADC = ∠ BAC. Show that CA² = CB.CD

Sides AB and AC and median AD of atriangle ABC are respectivelyproportional to sides PQ and PR andmedian PM of another triangle PQR.

Show that Δ ABC ~ Δ PQR

A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same timea 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 whereΔ ABC ~ Δ PQR, prove that

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

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

In Fig. 6.44, ABC and DBC are two triangles on thesame base BC. If AD intersects BC at O, show that

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

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

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

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

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

A. 2 : 1 B. 1 : 2

C. 4 : 1 D. 1 : 4

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

A. 2 : 3 B. 4 : 9

C. 81 : 16 D. 16 : 81

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.

(i) 7 cm, 24 cm, 25 cm

(ii) 3 cm, 8 cm, 6 cm

(iii) 50 cm, 80 cm, 100 cm

(iv) 13 cm, 12 cm, 5 cm

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

In Fig. 6.53, ABD is a triangle right angled at Aand AC ⊥ BD. Show that:

(i) AB² = BC . BD

(ii) AC² = BC . DC

(iii) AD² = BD . CD

ABC is an isosceles triangle right angled at C. Prove that

ABC is an isosceles triangle with AC = BC. If prove that ABC is a right triangle

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

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

In Fig. 6.54, O is a point in the interior of a triangleABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that

(i)

(ii)

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

A guy wire attached to a vertical pole of height 18 mis 24 m long and has a stake attached to the otherend. How far from the base of the pole should thestake be driven so that the wire will be taut?

An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At thesame time, another aeroplane leaves the same airport and flies due west at a speed of1200 km per hour. How far apart will be the two planes afterhours?

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

D and E are points on the sides CA and CBrespectively of a triangle ABC right angled at C.Prove that

The perpendicular from A on side BC of aΔABC intersects BC at D such that DB = 3 CD(see Fig. 6.55). Prove that

In an equilateral triangle ABC, D is a point on side BC such that Prove that 9AD² = 7AB²

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

Tick the correct answer and justify:

Inand the angle B is:

A. 120° B. 60°

C. 90° D. 45°

In Fig. 6.56, PS is the bisector of ∠QPR of Δ PQR. Prove that

In Fig. 6.57, D is a point on hypotenuse AC of Δ ABC, DM ⊥ BC and DN ⊥ AB. Prove that:

(i) (ii)

In Fig. 6.58, ABC is a triangle in which ∠ ABC > 90° and AD ⊥ CB produced. Prove that

In Fig. 6.59, ABC is a triangle in which ∠ ABC < 90° and AD ⊥ BC. Prove that

In Fig. 6.60, AD is a median of a triangle ABC and AM⊥ BC. Prove that:

(i)

(ii)

(iii)

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

In Fig. 6.61, two chords AB and CD intersect each other at the point P. Prove that:

(i) Δ APC ~ Δ DPB

(ii) AP . PB = CP . DP

In Fig. 6.62, two chords AB and CD of a circle intersect each other at the point P(when produced) outside the circle. Prove that

(i) Δ PAC ~ Δ PDB

(ii) PA . PB = PC . PD

In Fig. 6.63, D is a point on side BC of Δ ABC such thatProve that AD is thebisector of ∠ BAC

Nazima is fly fishing in a stream. The tip ofher fishing rod is 1.8 m above the surfaceof the water and the fly at the end of thestring rests on the water 3.6 m away and2.4 m from a point directly under the tip ofthe rod. Assuming that her string(from the tip of her rod to the fly) is taut,how much string does she have out(see Fig. 6.64)? If she pulls in the string atthe rate of 5 cm per second, what will bethe horizontal distance of the fly from herafter 12 seconds?