NCERT Solutions for class 10th Maths Chapter 6 Triangles

Exercise 6.1

Question 1. 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).

Sol.

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

Question 2. Give two different examples of pair of
(I) similar figures.
(ii) non-similar figures.

Sol.

(I) (1) Pair of equilateral triangles are similar figures.
(2) Pair of squares are similar figures
(ii) (1) A triangle and a quadrilateral form a pair of non-similar figures.
(2) A square and a rhombus form a pair of non-similar figure.

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

Exercise 6.2

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

Question 2. E and Fare points on the sides PQ and PR respectively of a PQR. For each of the following cases, state whether EF II 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 PE = 0.36 cm

Question 3. In figure, ifLM // CB and LN//CD,prove that(AM/AB = AN/AD)

Question 4. In figure, DE II AC and DF II AE. Prove that BF//FE = BE//EC


Question 5. In figure,DE II OQ and DF II OR.Show that EFIIQR.


Question 6. In figure A, Band Care points on OP, OQ and OR respectively such that AB II PQ and AC II PR. Show that BC II QR.


Question 7. 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).


Question 8. 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).


Question 9. ABCD is a trapezium in which AB II DC and its diagonals intersect each other at the point 0. Show that : AO/BO = CO/DO


Question 10. The diagonals of a quadrilateral ABCD intersect each other at the point 0 such that AO/BO = CO/DO .Show that ABCD is trapezium.

Exercise 6.3

Question 1. State which pairs of triangles in 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 :

Note: If two pairs of corresponding angles of two triangles are equal, then the third pair of corresponding angles are also equal.

Question 2. In figure, TRI ODC ~ TRI OBA, LBOC=125° and LCDO = 70°.Find LDOC, LDOC and LOAB.


Question 3. Diagonals AC and BD of a trapezium ABCD with AB II DC intersect each other at the point 0. Using a similarity criterion for two triangles, show that OA/OC=OB/OD

Question 4. In figure, QR/QS= QT/PR and Ll=L2.Show that TRI PQS~ TRI TQR.


Question 5. S and T are points on sides PR and QR of LiPQR such that LP= LRTS.Show that TRI RPQ ~ TRI RTS.

Question 6. In Fig. if AABE~=MCD. Prove that MOE~ AABC


Question 7. In Fig., altitudes AD and CE of AABC intersect each other at the point P. Show that:


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


Question 9. In Fig., ABC and AMP are two right triangles, right angled at B and M, respectively. Prove that:

Question 10. CD and GH are respectively the bisectors of LACB and LEGF such that D and H lie on sides AB and FE of ABC and EFG respectively. If C ~ FEG show that:


Question 11. In Fig., E is a point on side CB produced ofan isosceles triangle ABC with AB=AC.If AD per.BC and EFper.AC,prove that tri ABD ~ ECF

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 triangle PQR. Prove that ABC~PQR.

Question 13. ‘D’ is a point on the side BC of .:i ABC, such that LADC = LBAC. Prove that CA2 = CB x CD.

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 ABC ~PQR.

Question 15. 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 16. If AD and PM are medians of triangles ABC and PQR, respectively where ABC ~ PQR, prove that


Exercise 6.4

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

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

Question 3. In the given fig ABC and DBC are two triangles on the same base BC. If AD intersects BC at 0. Prove that

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

Question 5. D, E, Fare the mid points of sides BC, CA and AB respectively of ABC. Prove that

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.


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 diagonal.

Question 8. ABC and BOE are two equilateral triangles such that O is the mid-point of BC. Ratio of the areas of triangles ABC and BOE is
(a) 2 : 1 (b) 1 : 2
(c) 4 : 1 (d) 1 : 4


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


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

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

Question 3. In Fig., ABD is a triangle right angled at A and AC ..l BD. Show that
(i) AB2 = BC. BD (ii) AC2 = BC.DC (iii) AC2 = BO.CD

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


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

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

Note: In equilateral triangle, length of Median/Angle bisector/Altitude =3/2(side)

Question 7. 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 8. In Fig., O is point in the interior of a triangle ABC, OD per. BC, OE per.AC and OF per. AB. Show that

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.

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?

Question 11. An aeroplane leaves an airport and flies due north at speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after 1 1/2 hours ?

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

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.

Question 14. The perpendicular AD on the base BC of a ABC, intersects BC in D such that BD = 3CD. Prove that 2AB2 = 2AC2 +BC2

Question 15. In an equilateral triangle ABC, the side BC is trisected at ‘T’. Prove that 9AT2=7AB2 OR
In an equilateral .1ABC, D is a point such that BD = 3BC. Prove that
9AT2=7AB2.

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.

Question 17. Tick the correct answer and justify :
In .1ABC, AB = 6 3 cm, AC = 12 cm and BC= 6 cm. The angle B is :
(a) 120° (b) 60° (c) 90° (d) 45°


Exercise 6.6

Question 1. InFig.PS is the bisector of LQPR of LlPQR. Prove that QS/SR = PQ/PR

Note: This result is known as Angle bisector theorem.

Question 2. ABC is a right triangle with LABC = 90°, BD per AC, DM per BC and DN per. AB. Dis a point on hypotenuse AC of ABC. Prove that
(i) DM2 = DN x MC (ii) DN2 = DM x AN

Question 3. In Fig., ABC is a triangle in which LABC > 90° and AD per. CB produced. Prove that
AC2 = AB2 + BC2 + 2BC . BD

Question 4. In MBC, LB is an acute angle and AD .l BC, prove that AC2 = AB2+BC2-2BC.BD

Question 5. In Fig. AD is a median of a triangle ABC and AM per BC. Prove that:


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


Question 7. In Fig. two chords AB and CD intersect each other at the point P.Prove that :


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


Question 9. In Fig., D is a point on side BC of a ABC such that BD = AB . Prove that BD/CD=AB/AC is the bisector of LBAC.


Note: This is known as converse of Angle bisector theorem.

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 maway 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 (see Fig.)? 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?

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