Access NCERT Solutions for Mathematics Chapter 6- Triangles

                 EXERCISE 6.1

Question 1:

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

(i) All circles are__________. (congruent, similar)

(ii) All squares are________. (similar, congruent)

(iii) All___________triangles are (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)

Answer: 

(i) Similar

(ii) Similar

(iii) Equilateral

(iv) (a) Equal

(b) Proportional

Question 2: Give two different examples of pair of

(i) Similar figures (ii)Non-similar figures

Answer: (i) Two equilateral triangles with sides 1 cm and 2 cm

Question 3: State whether the following quadrilaterals are similar or not:

Answer:

Quadrilateral PQRS and ABCD are not similar as their corresponding sides are proportional, i.e. 1:2, but their corresponding angles are not equal.

            EXERCISE 6.2

Question 1: In figure.6.17. (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

(i)

Answer:

Let EC = x cm

It is given that DE || BC.

By using basic proportionality theorem, we obtain

Let AD = x cm

It is given that DE || BC.

By using basic proportionality theorem, we obtain

QUESTION 2: 

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.

(i) PE = 9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

(ii) PE = 4 cm, QE = 5 cm, PF = 8 cm and RF = 9 cm

(iii)PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm

Answer: 

Given that, PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm, FR = 2.4 cm

PE = 4 cm, QE = 4.5 cm, PF = 8 cm, RF = 9 cm

PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm, PF = 0.36 cm

Question 3: In the following figure, if LM || CB and LN || CD, prove that

In the given figure, LM || CB

By using basic proportionality theorem, we obtain

Question 4: In the following figure, DE || AC and DF || AE. Prove that

Question 5: In the following figure, DE || OQ and DF || OR, show that EF || QR.

Answer:

Question 6: In the following figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

Answer: 

Question 7: Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).

Answer:

Consider the given figure in which PQ is a line segment drawn through the mid-point P of line AB, such that 

Question 8: Using Converse of basic proportionality theorem, 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).

Answer: 

Consider the given figure in which PQ is a line segment joining the mid-points P and Q of line AB and AC respectively.

i.e., AP = PB and AQ = QC

It can be observed that

Question 9: ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that 

Answer: 

So, by using basic proportionality theorem, we obtain

Question 10:

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

Answer:

Let us consider the following figure for the given question.

Draw a line OE || AB

⇒ EO || DC [By the converse of basic proportionality theorem]

⇒ AB || OE || DC

⇒ AB || CD

∴ ABCD is a trapezium.

               EXERCISE 6.3

Question 1: State which pairs of triangles in the following figure 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:

(i)

(iv)

Answer:

(i) ∠A = ∠P = 60°

∠B = ∠Q = 80°

∠C = ∠R = 40°

Therefore, ∆ABC ∼ ∆PQR [By AAA similarity criterion]

(iii) The given triangles are not similar as the corresponding sides are not proportional.

(iv) The given triangles are not similar as the corresponding sides are not proportional.

(v) The given triangles are not similar as the corresponding sides are not proportional.

(vi) In ∆DEF,

∠D +∠E +∠F = 180º

(Sum of the measures of the angles of a triangle is 180º.)

70º + 80º +∠F = 180º

Similarly, in ∆PQR,

∠P +∠Q +∠R = 180º

(Sum of the measures of the angles of a triangle is 180º.)

∠P + 80º +30º = 180º

∠P = 70º

In ∆DEF and ∆PQR,

∠D = ∠P (Each 70°)

∠E = ∠Q (Each 80°)

∠F = ∠R (Each 30°)

∴ ∆DEF ∼ ∆PQR [By AAA similarity criterion]

Question 2: In the following figure, ∆ODC ∼ ∆OBA, ∠BOC = 125° and ∠CDO = 70°. Find ∠DOC,

∠DCO and ∠OAB

Answer:

DOB is a straight line.

∴ ∠DOC + ∠COB = 180°

➺ ∠DOC = 180° − 125°

= 55°

In ∆DOC,

∠DCO + ∠CDO + ∠DOC = 180°

(Sum of the measures of the angles of a triangle is 180º.)

➺ ∠DCO + 70º + 55º = 180°

➺ ∠DCO = 55°

It is given that ∆ODC ∼ ∆OBA.

∴ ∠OAB = ∠ OCD [Corresponding angles are equal in similar triangles.]

➺ ∠OAB = 55°

Question 3: 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 

Answer:

In ∆DOC and ∆BOA,

∠CDO = ∠ABO [Alternate interior angles as AB || CD]

∠DCO = ∠BAO [Alternate interior
angles as AB || CD]

∠DOC = ∠BOA [Vertically opposite
angles]

∴ ∆DOC ∼ ∆BOA [AAA similarity criterion]

Question 4: In the following figure, 

Answer:

Question 5: S and T are point on sides PR and QR of ∆PQR such that ∠P = ∠RTS. Show that

∆RPQ ∼ ∆RTS. 

Answer: 

In ∆RPQ and ∆RST,

∠RTS = ∠QPS (Given)

∠R = ∠R (Common angle)

∴ ∆RPQ ∼ ∆RTS (By AA similarity criterion)

Question 6: In the following figure, if ∆ABE ≅ ∆ACD, show that ∆ADE ∼ ∆ABC.

Answer:

It is given that ∆ABE ≅ ∆ACD.

∴ AB = AC [By CPCT] (1) And, AD = AE [By CPCT] (2)

In ∆ADE and ∆ABC,

[Dividing equation (2) by (1)]

∠A = ∠A [Common angle]

∴ ∆ADE ∼ ∆ABC [By SAS similarity criterion]

Question 7: In the following figure, 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

Answer:

In ∆AEP and ∆CDP,

∠AEP = ∠CDP (Each 90°)

∠APE = ∠CPD (Vertically opposite angles) Hence, by using AA similarity criterion,

∆AEP ∼ ∆CDP

(ii)

In ∆ABD and ∆CBE,

∠ADB = ∠CEB (Each 90°)

∠ABD = ∠CBE (Common)

Hence, by using AA similarity criterion,

∆ABD ∼ ∆CBE

(iii)

In ∆AEP and ∆ADB,

∠AEP = ∠ADB (Each 90°)

∠PAE = ∠DAB (Common)

Hence, by using AA similarity criterion,

∆AEP ∼ ∆ADB

(iv)

In ∆PDC and ∆BEC,

∠PDC = ∠BEC (Each 90°)

∠PCD = ∠BCE (Common angle) Hence, by using AA similarity criterion,

∆PDC ∼ ∆BEC

Question 8: E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ABE ∼ ∆CFB

Answer:

In ∆ABE and ∆CFB,

∠A = ∠C (Opposite angles of a parallelogram)

∠AEB = ∠CBF (Alternate interior angles as AE || BC)

∴ ∆ABE ∼ ∆CFB (By AA similarity criterion)

Question 9: In the following figure, ABC and AMP are two right triangles, right angled at B and M respectively, prove that:

(i) ∆ABC ∼ ∆AMP

Answer:

In ∆ABC and ∆AMP,

∠ABC = ∠AMP (Each 90°)

∠A = ∠A (Common)

∴ ∆ABC ∼ ∆AMP (By AA similarity criterion)

Question 10: CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ∆ABC and ∆EFG respectively. If ∆ABC ∼ ∆FEG, Show that:

(i)

(ii)∆DCB ∼ ∆HGE

(iii) ∆DCA ∼ ∆HGF 

Answer:

It is given that ∆ABC ∼ ∆FEG.

∴ ∠A = ∠F, ∠B = ∠E, and ∠ACB = ∠FGE

∠ACB = ∠FGE

∴ ∠ACD = ∠FGH (Angle bisector) And, ∠DCB = ∠HGE (Angle bisector) In ∆ACD and ∆FGH,

∠A = ∠F (Proved above)

∠ACD = ∠FGH (Proved above)

∴ ∆ACD ∼ ∆FGH (By AA similarity criterion)

In ∆DCB and ∆HGE,

∠DCB = ∠HGE (Proved above)

∠B = ∠E (Proved above)

∴ ∆DCB ∼ ∆HGE (By AA similarity criterion) In ∆DCA and ∆HGF,

∠ACD = ∠FGH (Proved above)

∠A = ∠F (Proved above)

∴ ∆DCA ∼ ∆HGF (By AA similarity criterion)

Question 11: In the following figure, E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If AD ⊥ BC and EF ⊥ AC, prove that ∆ABD ∼ ∆ECF

Answer:

It is given that ABC is an isosceles triangle.

∴ AB = AC

⇒ ∠ABD = ∠ECF

In ∆ABD and ∆ECF,

∠ADB = ∠EFC (Each 90°)

∠BAD = ∠CEF (Proved above)

∴ ∆ABD ∼ ∆ECF (By using AA similarity criterion)

Question 12: 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 the given figure). Show that ∆ABC ∼∆PQR.

Answer: 

In ∆ABD and ∆PQM,

∴ ∆ABD ∼ ∆PQM (By SSS similarity criterion)

⇒ ∠ABD = ∠PQM (Corresponding angles of similar triangles) In ∆ABC and ∆PQR,

∠ABD = ∠PQM (Proved above)

∴ ∆ABC ∼ ∆PQR (By SAS similarity criterion)

Question 13: D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that 

Answer: 

In ∆ADC and ∆BAC,

∠ADC = ∠BAC (Given)

∠ACD = ∠BCA (Common angle)

∴ ∆ADC ∼ ∆BAC (By AA similarity criterion)

We know that corresponding sides of similar triangles are in proportion.

Question 14: 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 

Answer:

Let us extend AD and PM up to point E and L respectively, such that AD = DE and PM

= ML. Then, join B to E, C to E, Q to L, and R to L.

We know that medians divide opposite sides. Therefore, BD = DC and QM = MR

Also, AD = DE (By construction) And, PM = ML (By construction)

In quadrilateral ABEC, diagonals AE and BC bisect each other at point D. Therefore, quadrilateral ABEC is a parallelogram.

∴ AC = BE and AB = EC (Opposite sides of a parallelogram are equal)

Similarly, we can prove that quadrilateral PQLR is a parallelogram and PR = QL, PQ

= LR

It was given that

∴ ∆ABE ∼ ∆PQL (By SSS similarity criterion)

We know that corresponding angles of similar triangles are equal.

∴ ∠BAE = ∠QPL … (1)

Similarly, it can be proved that ∆AEC ∼ ∆PLR and

∠CAE = ∠RPL … (2)

Adding equation (1) and (2), we obtain

∠BAE + ∠CAE = ∠QPL + ∠RPL

⇒ ∠CAB = ∠RPQ … (3)

In ∆ABC and ∆PQR,

∠CAB = ∠RPQ [Using equation (3)]

∴ ∆ABC ∼ ∆PQR (By SAS similarity criterion)

Question 15: A vertical pole of a length 6 m casts a shadow 4m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

Answer: 

Let AB and CD be a tower and a pole respectively.

Let the shadow of BE and DF be the shadow of AB and CD respectively.

At the same time, the light rays from the sun will fall on the tower and the pole at the same angle.

Therefore, ∠DCF = ∠BAE And, ∠DFC = ∠BEA

∠CDF = ∠ABE (Tower and pole are vertical to the ground)

∴ ∆ABE ∼ ∆CDF (AAA similarity criterion)

Therefore, the height of the tower will be 42 metres.

Question 16: If AD and PM are medians of triangles ABC and PQR, respectively where 

Answer:

It is given that ∆ABC ∼ ∆PQR

We know that the corresponding sides of similar triangles are in proportion.

                EXERCISE 6.4

Question 1: Let  and their areas be, respectively, 64 cm² and 121 cm². If EF = 15.4 cm, find BC.

Answer:

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

Answer:

Since AB || CD,

∴ ∠OAB = ∠OCD and ∠OBA = ∠ODC (Alternate interior angles) In ∆AOB and ∆COD,

∠AOB = ∠COD (Vertically opposite angles)

∠OAB = ∠OCD (Alternate interior angles)

∠OBA = ∠ODC (Alternate interior angles)

∴ ∆AOB ∼ ∆COD (By AAA similarity criterion)

Question 3: In the following figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that 

Answer: Let us draw two perpendiculars AP and DM on line BC

Question 4: If the areas of two similar triangles are equal, prove that they are congruent. 

Answer:

Let us assume two similar triangles as ∆ABC ∼ ∆PQR.

Question 5: D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ABC. Find the ratio of the area of ∆DEF and ∆ABC.

Answer:

D and E are the mid-points of ∆ABC.

Question 6: Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.

Answer: 

Let us assume two similar triangles as ∆ABC ∼ ∆PQR. Let AD and PS be the medians of these triangles.

Question 7: 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.

Answer:

Let ABCD be a square of side a.

Therefore, its diagonal 

Two desired equilateral triangles are formed as ∆ABE and ∆DBF.

Side of an equilateral triangle, ∆ABE, described on one of its sides = a

Side of an equilateral triangle, ∆DBF, described on one of its diagonals 

We know that equilateral triangles have all its angles as 60º and all its sides of the same length. Therefore, all equilateral triangles are similar to each other. Hence, the ratio between the areas of these triangles will be equal to the square of the ratio between the sides of these triangles.

Question 8: ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the area of triangles ABC and BDE is

(A) 2 : 1

(B) 1 : 2

(C) 4 : 1

(D) 1 : 4

Answer:

We know that equilateral triangles have all its angles as 60º and all its sides of the same length. Therefore, all equilateral triangles are similar to each other. Hence, the ratio between the areas of these triangles will be equal to the square of the ratio between the sides of these triangles.

Let side of ∆ABC = x

Therefore, side of 

Hence, the correct answer is (C).

Question 9: 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 

Answer:

If two triangles are similar to each other, then the ratio of the areas of these triangles will be equal to the square of the ratio of the corresponding sides of these triangles.

It is given that the sides are in the ratio 4:9.

Therefore, ratio between areas of these triangles = 

Hence, the correct answer is (D).

                  EXERCISE 6.5

Question 1: 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 

Answer:

(i) It is given that the sides of the triangle are 7 cm, 24 cm, and 25 Squaring the lengths of these sides, we will obtain 49, 576, and 625.

49 + 576 = 625

The sides of the given triangle are satisfying Pythagoras theorem. Therefore, it is a right triangle.

We know that the longest side of a right triangle is the hypotenuse. Therefore, the length of the hypotenuse of this triangle is 25 cm.

(ii) It is given that the sides of the triangle are 3 cm, 8 cm, and 6 Squaring the lengths of these sides, we will obtain 9, 64, and 36. However, 9 + 36 ≠ 64

Or, 3² + 6² ≠ 8²

Clearly, the sum of the squares of the lengths of two sides is not equal to the square of the length of the third side.

Therefore, the given triangle is not satisfying Pythagoras theorem. Hence, it is not a right triangle.

(iii) Given that sides are 50 cm, 80 cm, and 100

Squaring the lengths of these sides, we will obtain 2500, 6400, and 10000. However, 2500 + 6400 ≠ 10000

Or, 50² + 80² ≠ 100²

Clearly, the sum of the squares of the lengths of two sides is not equal to the square of the length of the third side.

Therefore, the given triangle is not satisfying Pythagoras theorem. Hence, it is not a right triangle.

(iv) Given that sides are 13 cm, 12 cm, and 5

Squaring the lengths of these sides, we will obtain 169, 144, and 25.

Clearly, 144 +25 = 169

The sides of the given triangle are satisfying Pythagoras theorem. Therefore, it is a right triangle.

We know that the longest side of a right triangle is the hypotenuse. Therefore, the length of the hypotenuse of this triangle is 13 cm.

Question 2: PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM² = QM × MR.

Answer: 

Question 3: In the following figure, ABD is a triangle right angled at A and AC ⊥ BD. Show that

(i) AB² = BC × BD

(ii) AC² = BC × DC

(iii) AD² = BD × CD

Answer:

(i) In 

∠DCA = ∠ DAB (Each 90º)

∠CDA = ∠ ADB (Common angle)

Question 4: ABC is an isosceles triangle right angled at C. prove that AB² = 2 AC². 

Answer:

Given that ∆ABC is an isosceles triangle.

∴ AC = CB

Applying Pythagoras theorem in ∆ABC (i.e., right-angled at point C), we obtain

Question 5: ABC is an isosceles triangle with AC = BC. If AB² = 2 AC², prove that ABC is a right triangle.

Answer:

Question 6: ABC is an equilateral triangle of side 2a. Find each of its altitudes. 

Answer:

Let AD be the altitude in the given equilateral triangle, ∆ABC. We know that altitude bisects the opposite side.

∴ BD = DC = a

In an equilateral triangle, all the altitudes are equal in length.

Therefore, the length of each altitude will be 

Question 7: Prove that the sum of the squares of the sides of rhombus is equal to the sum of the squares of its diagonals.

Answer: 

In ∆AOB, ∆BOC, ∆COD, ∆AOD,

Applying Pythagoras theorem, we obtain

Question 8: In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥AC and OF ⊥ AB. Show that

(i) OA² + OB² + OC² − OD² − OE² − OF² = AF² + BD² + CE²

(ii) AF² + BD² + CE² = AE² + CD² + BF²

Answer:

Join OA, OB, and OC.

Question 9: 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.

Answer: 

Let OA be the wall and AB be the ladder.

Therefore, by Pythagoras theorem,

Therefore, the distance of the foot of the ladder from the base of the wall is 6 m.

Question 10 : 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?

Answer: 

Let OB be the pole and AB be the wire. By Pythagoras theorem,

Question 11: An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after 

Answer:  

Let these distances be represented by OA and OB respectively. Applying Pythagoras theorem

Question 12: 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.

Answer:

Let CD and AB be the poles of height 11 m and 6 m. Therefore, CP = 11 − 6 = 5 m

From the figure, it can be observed that AP = 12m Applying Pythagoras theorem for ∆APC, we obtain

Therefore, the distance between their tops is 13 m.

Question 13: 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

Answer:

Question 14: The perpendicular from A on side BC of a ∆ABC intersect BC at D such that DB = 3 CD. Prove that 2 AB² = 2 AC² + BC²

Answer:

Applying Pythagoras theorem for ∆ACD, we obtain

Applying Pythagoras theorem in ∆ABD, we obtain

Question 15: In an equilateral triangle ABC, D is a point on side BC such that BD =  BC. Prove that 9 AD² = 7 AB².

Answer:

Let the side of the equilateral triangle be a, and AE be the altitude of ∆ABC.

Applying Pythagoras theorem in ∆ADE, we obtain AD² = AE² + DE²

⇒ 9 AD² = 7 AB²

Question 16: 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.

Answer:

Let the side of the equilateral triangle be a, and AE be the altitude of ∆ABC.

∴ BE = EC = 

Applying Pythagoras theorem in ∆ABE, we obtain AB² = AE² + BE²

4AE² = 3a²

⇒ 4 × (Square of altitude) = 3 × (Square of one side)

Question 17: Tick the correct answer and justify: In ∆ABC, AB = cm, AC = 12 cm and BC = 6 cm.

The angle B is:

(A) 120°

(B) 60°

(C) 90° (D) 45°

Answer:

It can be observed that

AB² = 108

AC² = 144 And, BC² = 36

AB² +BC² = AC²

The given triangle, ∆ABC, is satisfying Pythagoras theorem.

Therefore, the triangle is a right triangle, right-angled at B.

∴ ∠B = 90°

Hence, the correct answer is (C).

               EXERCISE 6.6

Question 1: In the given figure, PS is the bisector of ∠QPR of ∆PQR. Prove that    

Answer:

Let us draw a line segment RT parallel to SP which intersects extended line segment QP at point T.

Given that, PS is the angle bisector of ∠QPR.

∠QPS = ∠SPR … (1)

By construction,

∠SPR = ∠PRT (As PS || TR) … (2)

∠QPS = ∠QTR (As PS || TR) … (3)

Using these equations, we obtain

∠PRT = ∠QTR

∴ PT = PR

By construction, PS || TR

By using basic proportionality theorem for ∆QTR,

Question 2: In the given figure, D is a point on hypotenuse AC of ∆ABC, DM ⊥ BC and DN ⊥ AB, Prove that:

(i) DM² = MC

(ii) DN² = AN

Answer:

(i) Let us join DB.

We have, DN || CB, DM || AB, and ∠B = 90°

∴ DMBN is a rectangle.

∴ DN = MB and DM = NB

The condition to be proved is the case when D is the foot of the perpendicular drawn from B to AC.

∴ ∠CDB = 90°

⇒ ∠2 + ∠3 = 90° … (1) In ∆CDM,

∠1 + ∠2 + ∠DMC = 180°

⇒ ∠1 + ∠2 = 90° … (2) In ∆DMB,

∠3 + ∠DMB + ∠4 = 180°

⇒ ∠3 + ∠4 = 90° … (3)

From equation (1) and (2), we obtain

∠1 = ∠3

From equation (1) and (3), we obtain

∠2 = ∠4

In ∆DCM and ∆BDM,

∠1 = ∠3 (Proved above)

∠2 = ∠4 (Proved above)

∴ ∆DCM ∼ ∆BDM (AA similarity criterion)

DM² = DN × MC

(ii) In right triangle DBN,

∠5 + ∠7 = 90° … (4)

In right triangle DAN,

∠6 + ∠8 = 90° … (5)

D is the foot of the perpendicular drawn from B to AC.

∴ ∠ADB = 90°

⇒ ∠5 + ∠6 = 90° … (6)

From equation (4) and (6), we obtain

∠6 = ∠7

From equation (5) and (6), we obtain

∠8 = ∠5

In ∆DNA and ∆BND,

∠6 = ∠7 (Proved above)

∠8 = ∠5 (Proved above)

∴ ∆DNA ∼ ∆BND (AA similarity criterion)

DN² = AN × NB

⇒ DN² = AN × DM  (As NB = DM)

Question 3: In the given figure, ABC is a triangle in which ∠ABC> 90° and AD ⊥ CB produced. Prove that AC² = AB² + BC² + 2BC.BD.

Answer:

Applying Pythagoras theorem in ∆ADB, we obtain AB² = AD² + DB² … (1)

Applying Pythagoras theorem in ∆ACD, we obtain AC² = AD² + DC²

AC² = AD² + (DB + BC)²

AC² = AD² + DB² + BC² + 2DB × BC

AC² = AB² + BC² + 2DB × BC [Using equation (1)]

Question 4: In the given figure, ABC is a triangle in which ∠ABC < 90° and AD ⊥ BC. Prove that AC² = AB² + BC² − 2BC.BD.

Answer:

Applying Pythagoras theorem in ∆ADB, we obtain AD² + DB² = AB²

⇒ AD² = AB² − DB² … (1)

Applying Pythagoras theorem in ∆ADC, we obtain AD² + DC² = AC²

AB² − BD² + DC² = AC² [Using equation (1)] AB² − BD² + (BC − BD)² = AC²

AC² = AB² − BD² + BC² + BD² −2BC × BD

= AB² + BC² − 2BC × BD

Question 5: In the given figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that:

Answer:

(i) Applying Pythagoras theorem in ∆AMD, we obtain AM² + MD² = AD² … (1)

Applying Pythagoras theorem in ∆AMC, we obtain AM² + MC² = AC²

AM² + (MD + DC)² = AC²

(AM² + MD²) + DC² + 2MD.DC = AC²

AD² + DC² + 2MD.DC = AC² [Using equation (1)]

(ii) Applying Pythagoras theorem in ∆ABM, we obtain AB² = AM² + MB²

= (AD² − DM²) + MB²

= (AD² − DM²) + (BD − MD)²

= AD² − DM² + BD² + MD² − 2BD × MD

= AD² + BD² − 2BD × MD

(iii) Applying Pythagoras theorem in ∆ABM, we obtain AM² + MB² = AB² … (1)

Applying Pythagoras theorem in ∆AMC, we obtain AM² + MC² = AC² … (2)

Adding equations (1) and (2), we obtain 2AM² + MB² + MC² = AB² + AC²

2AM² + (BD − DM)² + (MD + DC)² = AB² + AC²

2AM²+BD² + DM² − 2BD.DM + MD² + DC² + 2MD.DC = AB² + AC²

2AM² + 2MD² + BD² + DC² + 2MD (− BD + DC) = AB² + AC²

Question 6: Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

Answer:

Let ABCD be a parallelogram.

Let us draw perpendicular DE on extended side AB, and AF on side DC. Applying Pythagoras theorem in ∆DEA, we obtain

DE² + EA² = DA² … (i)

Applying Pythagoras theorem in ∆DEB, we obtain DE² + EB² = DB²

DE² + (EA + AB)² = DB²

(DE² + EA²) + AB² + 2EA × AB = DB² DA² + AB² + 2EA × AB = DB² … (ii)

Applying Pythagoras theorem in ∆ADF, we obtain AD² = AF² + FD²

Applying Pythagoras theorem in ∆AFC, we obtain AC² = AF² + FC²

= AF² + (DC − FD)²

= AF² + DC² + FD² − 2DC × FD

= (AF² + FD²) + DC² − 2DC × FD AC² = AD² + DC² − 2DC × FD … (iii)

Since ABCD is a parallelogram, AB = CD … (iv)

And, BC = AD … (v) In ∆DEA and ∆ADF,

∠DEA = ∠AFD (Both 90°)

∠EAD = ∠ADF (EA || DF)

AD = AD (Common)

∴ ∆EAD    ∆FDA (AAS congruence criterion)

⇒ EA = DF … (vi)

Adding equations (i) and (iii), we obtain

DA² + AB² + 2EA × AB + AD² + DC² − 2DC × FD = DB² + AC² DA² + AB² + AD² + DC² + 2EA × AB − 2DC × FD = DB² + AC² BC² + AB² + AD² + DC² + 2EA × AB − 2AB × EA = DB² + AC²

[Using equations (iv) and (vi)]

AB² + BC² + CD² + DA² = AC² + BD²

Question 7: In the given figure, two chords AB and CD intersect each other at the point P. prove that:

(i) ∆APC ∼ ∆DPB

(ii) AP.BP = CP.DP

Answer:

Let us join CB.

(i)  In ∆APC and ∆DPB,

∠APC = ∠DPB (Vertically opposite angles)

∠CAP = ∠BDP (Angles in the same segment
for chord CB)

∆APC ∼ ∆DPB (By AA similarity criterion)

(ii)  We have already proved that

∆APC ∼ ∆DPB

We know that the corresponding sides of similar triangles are proportional.

Question 8: In the given figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that

(i) ∆PAC ∼ ∆PDB

(ii) PB = PC.PD

Answer: 

(i) In ∆PAC and ∆PDB,

∠P = ∠P (Common)

∠PAC = ∠PDB (Exterior angle of a cyclic quadrilateral is ∠PCA = ∠PBD equal to the opposite interior angle)

∴ ∆PAC ∼ ∆PDB

(ii) We know that the corresponding sides of similar triangles are

Question 9: In the given figure, D is a point on side BC of ∆ABC such that . Prove that AD is the bisector of ∠BAC.

Answer: Let us extend BA to P such that AP = AC. Join PC.

By using the converse of basic proportionality theorem, we obtain AD || PC

⇒ ∠BAD = ∠APC (Corresponding angles) … (1)

And, ∠DAC = ∠ACP (Alternate interior angles) … (2)

By construction, we have AP = AC

⇒ ∠APC = ∠ACP … (3)

On comparing equations (1), (2), and (3), we obtain

∠BAD = ∠APC

⇒ AD is the bisector of the angle BAC

Question 10 :

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, ho much string does she have out (see Fig. 6.64)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?

Answer:

Let AB be the height of the tip of the fishing rod from the water surface. Let BC be the horizontal distance of the fly from the tip of the fishing rod.

Then, AC is the length of the string.

AC can be found by applying Pythagoras theorem in ∆ABC. AC² = AB² + BC²

AB² = (1.8 m)² + (2.4 m)² 

AB² = (3.24 + 5.76) m² 

AB² = 9.00 m²

Thus, the length of the string out is 3 m.

She pulls the string at the rate of 5 cm per second.

Therefore, string pulled in 12 seconds = 12 × 5 = 60 cm = 0.6 m

Let the fly be at point D after 12 seconds. Length of string out after 12 seconds is AD.

AD = AC − String pulled by Nazima in 12 seconds

= (3.00 − 0.6) m

= 2.4 m In ∆ADB,

AB² + BD² = AD²

(1.8 m)² + BD² = (2.4 m)²

BD² = (5.76 − 3.24) m² = 2.52 m² BD = 1.587 m

Horizontal distance of fly = BD + 1.2 m

= (1.587 + 1.2) m

= 2.787 m

= 2.79 m