NCERT Exemplar Class 10 Maths Solutions Chapter 6 Triangles

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NCERT Exemplar Problems Class 10 Maths Solutions Chapter 6 Triangles

Exercise 6.1 Multiple Choice Questions (MCQs)

Question 1:
In figure, if ∠BAC =90° and AD⊥BC. Then,

(a) BD.CD = BC²                         (b) AB.AC = BC²                        (c) BD.CD=AD²                          (d) AB.AC =AD²
Solution:
(c) In ΔADB and ΔADC,

Question 2:
If the lengths of the diagonals of rhombus are 16 cm and 12 cm. Then, the length of the sides of the rhombus is
(a) 9 cm                     (b) 10 cm                  (c) 8 cm                     (d) 20 cm
Solution:
(b) We know that, the diagonals of a rhombus are perpendicular bisector of each other.
Given,                             AC = 16 cm and BD = 12 cm                                                [let]
∴                                       AO = 8cm, SO = 6cm
and                           ∠AOB = 90°
In right angled ∠AOB,

Question 3:
If ΔABC ~ ΔEDF and ΔABC is not similar to ΔDEF, then which of the following is not true?
(a) BC · EF = AC · FD                                 (b) AB · EF = AC · DE
(c) BC · DE = AB · EF                                 (d) BC · DE = AB · FD
Solution:

Question 4:
If in two Δ PQR ,\(\frac { AB }{ QR } =\frac { BC }{ PR } =\frac { CA }{ PQ } \),then
(a)Δ PQR~Δ CAB                                       (b) Δ PQR ~ Δ ABC
(c)Δ CBA ~ Δ PQR                                     (d) Δ BCA ~ Δ PQR
Solution:
(a) Given, in two Δ ABC and Δ PQR,\(\frac { AB }{ QR } =\frac { BC }{ PR } =\frac { CA }{ PQ } \)
which shows that sides of one triangle are proportional to the side of the other triangle, then their corresponding angles are also equal, so by SSS similarity, triangles are similar.
i.e.,                                           Δ CAB ∼ Δ PQR

Question 5:
In figure, two line segments AC and BD intersect each other at the point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, ∠APB = 50° and ∠CDP = 30°. Then, ∠PBA is equal to
(a) 50°                           (b) 30°                              (c) 60°                               (d) 100°
Solution:

Question 6:
If in two Δ DEF and Δ PQR,∠D =∠Q and ∠R = ∠E,then which of the following is not true?

Solution:
(b) Given,in ΔDEF,∠D =∠Q,∠R = ∠E

Question 7:
In Δ ABC and ΔDEF, ∠B = ∠E, ∠F = ∠C and AB = 30E Then, the two triangles are
(a) congruent but not similar                   (b) similar but not congruent
(c) neither congruent nor similar            (d) congruent as well as similar
Solution:
(b) In ΔABC and ΔDEF, ∠B = ∠E, ∠F = ∠C and AB = 3DE

We know that, if in two triangles corresponding two angles are same, then they are similar by AAA similarity criterion. Also, ΔA8C and ΔDEF do not satisfy any rule of congruency, (SAS, ASA, SSS), so both are not congruent.

Question 8:

Solution:

Question 9:
If ΔABC ~ΔDFE, ∠A = 30°, ∠C = 50°, AB = 5 cm, AC = 8 cm and OF = 7.5 cm. Then, which of the following is true?
(a) DE =12 cm, ∠F =50°                            (b) DE = 12 cm, ∠F =100°
(c) EF = 12 cm, ∠D = 100°                        (d) EF = 12 cm,∠D =30°
Solution:
(b) Given, AABC ~ ADFE, then ∠A = ∠D = 30°, ∠C = ∠E = 50°

Question 10:
If in ΔABC and ΔDEF, \(\frac { AB }{ DE } =\frac { BC }{ FD } \), then they will be similar, when
(a) ∠B = ∠E                                               (b) ∠A = ∠D
(c)∠B = ∠D                                                (d) ∠A = ∠F
Solution:
(c) Given, in ΔABC and ΔEDF,

Question 11:

(a) 10 cm                  (b) 12 cm                 (c) \(\frac { 20 }{ 3 } \)cm                    (d) 8 cm
Solution:
(a) Given, Δ ABC ~Δ QRP, AB = 18cm and BC = 15cm

We know that, the ratio of area of two similar triangles is equal to the ratio of square of their corresponding sides.

Question 12:
If S is a point on side PQ of a Δ PQR such that PS = QS = RS, then

Solution:

Exercise 6.2 Very Short Answer Type Questions

Question 1:
Is the triangle with sides 25 cm, 5 cm and 24 cm a right triangle? Give reason for your answer.
Solution:
False
Let a = 25 cm, b = 5 cm and c =24 cm
Now,                                             b² + c² = (5)² + (24)²
                                                                     = 25+ 576 = 601 ≠ (25)²
Hence, given sides do not make a right triangle because it does not satisfy the property of Pythagoras theorem.

Question 2:
It is given that ΔDEF ~ ΔRPQ. Is it true to say that ∠D = ∠R and ∠F = ∠P ? Why?
Solution:
False
We know that, if two triangles are similar, then their corresponding angles are equal.
∴                                       ∠D = ∠R, ∠E = ∠P and ∠F = Q

Question 3:
A and B are respectively the points on the sides PQ and PR of a ΔPQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB || QR ? Give reason for your answer.
Solution:
False
Given, PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm

Question 4:
In figure, BD and CE intersect each other at the point P. Is ΔPBC ~ ΔPDE? Why?

Solution:
True

Since, one angle of ΔPBC is equal to one angle of ΔPDE and the sides including these angles are proportional, so both triangles are similar.
Hence, ΔPBC ∼ ΔPDE, by SAS similarity criterion.

Question 5:
In ΔPQR and ΔMST, ∠P = 55°, ∠Q =25°, ∠M = 100° and ∠S = 25°. Is ΔQPR ~ ΔTSM? Why?
Solution:
False
We know that, the sum of three angles of a triangle is 180°.
In ΔPQR,                                     ∠P + ∠Q + ∠R = 180°
⇒                                              55° + 25° + ∠R = 180°
⇒                                   ∠R = 180° – (55° + 25°)= 180° – 80° =100°
In ΔTSM,                                    ∠T + ∠S + ∠M = 180°
⇒                                               ∠T + ∠25°+ 100° = 180°
⇒                                                                    ∠T = 180°-(25° +100°)
=180°-125°= 55°

In ΔPQR and A TSM, and
∠P = ∠T, ∠Q = ∠S,
and                                                        ∠R = ∠M
ΔPQR ~ ΔTSM [since, all corresponding angles are equal]
Hence, Δ QPR is not similar to ΔTSM, since correct correspondence is P ↔ T, Q < r→ S and R ↔M

Question 6:
Is the following statement true? Why? “Two quadrilaterals are similar, if their corresponding angles are equal”.
Solution:
False
Two quadrilaterals are similar, if their corresponding angles are equal and corresponding sides must also be proportional.

Question 7:
Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Are the two triangles similar? Why?
Solution:
True
Here, the corresponding two sides and the perimeters of two triangles are proportional, then third side of both triangles will also in proportion.

Question 8:
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 two triangles will be similar? Why?
Solution:
True
Let two right angled triangles be ΔABC and ΔPQR.

Question 9:
The ratio of the corresponding altitudes of two similar triangles is \(\frac { 3 }{ 5 } \). Is it
correct to say that ratio of their areas is \(\frac { 6 }{ 5 } \) ? Why?
Solution:
False
By the property of area of two similar triangles,

So, given statement is not correct,

Question 10:
D is a point on side QR of ΔPQR such that PD ⊥ QR. Will it be correct to say that ΔPQD ~ ΔRPD? Why?
Solution:
False
In ΔPQD and ΔRPD,
PD = PD                                                 [common side]
∠PDQ = ∠PDR                                                    [each 90°]

Here, no other sides or angles are equal, so we can say that ∠PQD is not similar to ΔRPD. But, if ∠P = 90°,
then ∠DPQ = ∠PRD
[each equal to 90° – ∠0 and by ASA similarity criterion, ΔPQD ~ΔRPD]

Question 11:
In figure, if ∠D = ∠C, then it is true that ΔADE ~ ΔACB? Why?

Solution:
True
In ΔADE and ΔACB,
∠A = ∠A                                                 [common angle]
∠D = ∠C                                                                 [given]
ΔADE ~ ΔACB                     [by AAA similarity criterion]

Question 12:
Is it true to say that, if in two triangles, an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle, then the triangles are similar? Give reason for your answer.
Solution:
False
Because, according to SAS similarity criterion, if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
Here, one angle and two sides of two triangles are equal but these sides not including equal angle, so given statement is not correct.

Exercise 6.3 Short Answer Type Questions

Question 1:
In a ΔPQR, PR² – PQ² = QR² and M is a point on side PR such that QM ⊥ PR.
Prove that QM² =PM × MR.
Solution:
Given In A PQR,                             PR² – PQ² = QR² and QM ⊥ PR
To prove                                         QM² = PM x MR
Proof Since,                        PR² – PQ² = QR²
⇒                                                     PR² = PQ² + QR²

Question 2:
Find the value of x for which DE \(\parallel \) AB in given figure.
Solution:

Question 3:
In figure, if ∠1 =∠2 and ΔNSQ = ΔMTR, then prove that ΔPTS ~ ΔPRQ.

Solution:

Question 4:
Diagonals of a trapezium PQRS intersect each other at the point 0, PQ || RS and PQ = 3 RS. Find the ratio of the areas of
Δ POQ and Δ ROS.
Solution:
Given PQRS is a trapezium in which PQ || PS and PQ = 3 RS


Hence, the required ratio is 9 :1.

Question 5:
In figure, if AB || DC and AC, PQ intersect each other at the point 0. Prove that OA . CQ = 0C . AP.

Solution:
Given AC and PQ intersect each other at the point O and AB || DC
Prove that OA . CQ = 0C . AP.

Question 6:
Find the altitude of an equilateral triangle of side 8 cm.
Solution:
Let ABC be an equilateral triangle of side 8 cm i.e., AB = BC = CA = 8 cm. Draw altitude AD which is perpendicular to BC. Then, D is the mid-point of BC.

Question 7:
If ΔABC ~ ΔDEF, AB = 4 cm, DE = 6, EF = 9 cm and FD = 12 cm, then find the perimeter of Δ ABC.
Solution:
Given AB = 4cm, DE = 6cm and EF = 9cm and FD = 12 cm

Question 8:
In figure, if DE || BC, then find the ratio of ar (Δ ADE) and ar (DECB).

Solution:

Question 9:
ABCD is a trapezium in which AB || DC and P,Q are points on AD and BC respectively, such that PQ || DC, if PD = 18 cm, BQ = 35 cm and QC = 15 cm, find AD.
Solution:
Given, a trapezium ABCD in which AB || DC. P and Q are points on AD and BC, respectively such that PQ || DC. Thus, AB || PQ || DC.

Question 10:
Corresponding sides of two similar triangles are in the ratio of 2 : 3. If the area of the smaller triangle is 48 cm², then find the area of the larger triangle.
Solution:
Given, ratio of corresponding sides of two similar triangles = 2:3 or \(\frac { 2 }{ 3 } \)
Area of smaller triangle = 48 cm²
By the property of area of two similar triangle,
Ratio of area of both riangles = (Ratio of their corresponding sides)²

Question 11:
In a Δ PQR, N is a point on PR, such that QN ⊥ PR. If PN . NR = QN², then prove that ∠PQR = 90°.
Solution:
Given ΔPQR, N is a point on PR, such that QN ⊥ PR

Question 12:
Areas of two similar triangles are 36 cm² and 100 cm². If the length of a side of the larger triangle is 20 cm. Find the length of the corresponding side of the smaller triangle.
Solution:
Given, area of smaller triangle = 36 cm² and area of larger triangle = 100 cm²
Also, length of a side of the larger triangle = 20 cm
Let length of the corresponding side of the smaller triangle = x cm
By property of area of similar triangle,

Hence, the length of corresponding side of the smaller triangle is 12 cm.

Question 13:
In given figure, if ∠ACB = ∠CDA, AC = 8 cm and AD = 3 cm, then find BD.
Solution:
Given, AC = 8 cm, AD = 3cm and            ∠ACB = ∠CDA
From figure,                                                ∠CDA = 90°
∠ACB = ∠CDA = 90°

Question 14:
A 15 high tower casts a shadow 24 Long at a certain time and at the same time, a telephone pole casts a shadow 16 long. Find the height of the telephone pole.
Solution:
Let BC = 15 m be the tower and its shadow AB is 24 m. At that time ∠CAB = 8, Again, let EF = h be a telephone pole and its shadow DE = 16 m. At the same time ∠EDF = 8 Here, ΔASC and ΔDEF both are right angled triangles.


Hence, the height of the telephone pole is 10 m.

Question 15:
Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches.
Solution:
Let AB be a vertical wall and AC = 10 m is a ladder. The top of the ladder reaches to A and distance of ladder from the base of the wall BC is 6 m.

Hence, the height of the point on the wall where the top of the ladder reaches is 8 cm.

Exercise 6.4 Long Answer Type  Questions

Question 1:
In given figure, if ∠A = ∠C, AB = 6 cm, BP = 15 cm, AP = 12 cm and CP = 4 cm, then find the lengths of PD and CD.

Solution:
Given,∠A = ∠C, AS = 6cm, BP = 15cm, AP = 12 cm and CP = 4cm
In ΔAPB and ΔCPD,                       ∠A =∠C                                                                    [given]
∠APS = ∠CPD                              [vertically opposite angles]

Hence, length of PD = 5 cm and length of CD = 2 cm

Question 2:
It is given that ΔABC ~ ΔEDF such that AB = 5 cm, AC = 7 cm, DF = 15 cm and DE = 12 cm. Find the lengths of the remaining sides of the triangles,
Solution:
Given, ΔABC ~ ΔEDF, so the corresponding sides of ΔASC and ΔEDF are in the same ratio.

Hence, lengths of the remaining sides of the triangles are EF = 16.8 cm and SC = 625 cm.

Question 3:
Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio.
Solution:
Let a ΔABC in which a line DE parallel to SC intersects AB at D and AC at E.To prove DE divides the two sides in the same ratio.

Question 4:
In the given figure, if PQRS is a parallelogram and AB || PS, then prove that 0C || SR.
Solution:
Given PQRS is a parallelogram, so PQ || SR and PS || QR. Also, AB || PS^.

Question 5:
A 5m long ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4 m high. If the foot of the ladder is moved 1.6 m towards the wall, then find the distance by which the top of the ladder would slide upwards on the wall.
Solution:
Let AC be the ladder of length 5 m and BC = 4 m be the height of the wall, which ladder is placed. If the foot of the ladder is moved 1.6 m towards the wall i.e, AD = 1.6 m, then the ladder is slide upward i.e. ,CE = x m.
In right angled ΔABC,

Hence, the top of the ladder would slide upwards on the wall at distance 0.8 m.

Question 6:
For going to a city B from city A there is a route via city C such that AC ⊥ CB, AC = 2x km and CB = 2(x+ 7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.
Solution:
Given, AC ⊥ CB, km,CB = 2(x + 7) km and AB = 26 km
On drawing the figure, we get the right angled Δ ACB right angled at C.
Now, In ΔACB, by Pythagoras theorem,

Hence, the required saved distance is 34 – 26  i.e., 8 km.

Question 7:
A flag pole 18 m high casts a shadow 9.6 m long. Find the distance of the top of the pole from the far end of the shadow.
Solution:
Let BC = 18 m be the flag pole and its shadow be AB = 9.6 m. The distance of the top of the pole, C from the far end i.e., A of the shadow is AC.


Hence, the required distance is 20.4 m.

Question 8:
A street light bulb is fixed on a pole 6 m above the level of the street. If a woman of height 1.5 m casts a shadow of 3 m, then find how far she is away from the base of the pole.
Solution:
Let A be the position of the street bulb fixed on a pole AB = 6 m and CD = 1.5 m be the height of a woman and her shadow be ED = 3 m. Let distance between pole and woman be x m.

Hence, she is at the distance of 9 m from the base of the pole.

Question 9:
In given figure, ABC is a triangle right angled at B and BD ⊥ AC. If AD = 4 cm and CD = 5 cm, then find BD and AB.

Solution:

Question 10:
In given figure PQR is a right triangle, right angled at Q and QS ⊥ PR. If PQ = 6 cm and PS = 4 cm, then find QS, RS and QR.

Solution:
Given, ΔPQR in which ∠Q = 90°, QS ⊥ PR and PQ = 6 cm, PS = 4 cm In ΔSQP and ΔSRQ,

Question 11:
In ΔPQR, PD ⊥ QR such that D lies on QR, if PQ = a, PR = b, QD = c and DR = d, then prove that (a + b)(a -b) = (c + d) (c -d).
Solution:
Given In A PQR, PD 1 QR, PQ = a, PR = b,QD = c and DR =d
To prove (a + b) (a-b) = (c + d)(c-d)
Proof In right angled ΔPDQ,

Question 12:
In a quadrilateral ΔBCD, ∠A+ ∠D = 90°. Prove that
AC² + BD² = AD² + BC².
Solution:
Given Quadrilateral ΔBCD, in which ∠A+ ∠D = 90°
To prove                                  AC² + BD² = AD² + BC²
Construct Produce AB and CD to meet at E.

Question 13:

Solution:

Question 14:
In figure, PA, QB, RC and SD are all perpendiculars to a line i, AB = 6 cm, BC = 9 cm, CD = 12 cm and SP = 36 cm. Find PQ, QR and RS.

Solution:
Given, AS = 6 cm, BC = 9 cm, CD = 12 cm and SP = 36 cm
Also, PA, QB, RC and SD are all perpendiculars to line l.
PA || QS|| SC || SD
By basic proportionality theorem,

Question 15:
0 is the point of intersection of the diagonals AC and BD of a trapezium ABCD with AB || DC. Through 0, a line segment PQ is drawn parallel to AB meeting AD in P and BC in Q, prove that PO = QO.
Solution:
Given ABCD is a trapezium. Diagonals AC and BD are intersect at 0.
PQ||AB||DC.     
      

Question 16:
In figure, line segment DF intersects the side AC of a ΔABC at the point E such that E is the mid-point of CA and
∠AEF = ∠AFE. Prove that  \(\frac { BD }{ CD } =\frac { BF }{ CF } \).

Solution:

Question 17:
Prove that the area of the semi-circle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the semi-circles drawn on the other two sides of the triangle.
Solution:
Let ABC be a right triangle, right angled at B and AB = y, BC = x.
Three semi-circles are drawn on the sides AB, BC and AC, respectively with diameters AB, BC and AC, respectively.
Again, let area of circles with diameters AB, BC and AC are respectively A₁, A₂ and A₃.

Question 18:
Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangle drawn on the other two sides of the triangle.
Solution:
Let a right triangle BAC in which ∠A is right angle and AC = y, AB = x.
Three equilateral triangles ΔAEC, Δ AFB and ΔCBD are drawn on the three sides of ΔABC. Again let area of triangles made on AC, AS and BC are A₁, A₂ and A₃, respectively.
To prove A₃ = A₁ + A₂

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