Find below the NCERT Exemplar problems and their solutions for Class 10 Mathematics Chapter, Triangles:

Exercise 6.4

Long Answer Type Questions (Q. NO. 1-6):

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

Question. 2 It is given that DABC ~ DEDF such that AB = 5 cm, AC = 7cm, DF = 15cm and DE = 12cm. Find the lengths of the remaining sides of the triangles.

Solution.

Hence, the lengths of remaining sides of given triangles are : EF = 16.8cm and BC = 6.25cm

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  DABC in which a line DE is drawn parallel to BC intersecting AB at D and AC at E.

Hence proved.

Question. 4 In the given figure, if PQRS is a parallelogram and AB || PS then prove that OC || SR.

Solution.

Question. 5 A 5m long Ladder is placed leaning towards a vertical wall such that it reaches the wall at a point 4m high. If the foot of the ladder is moved 1.6m 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 and BC be the wall.

Let the two positions of ladder be represented by AC and DE.

Hence, the top of the ladder would slide upwards on the wall by a distance equal to 0.8m.

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 26km 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.        

Using Pythagoras theorem in right angled DACB, we have:

            AB² = AC² + BC²

⟹        (26)² = (2x)² + {2(x + 7)²

⟹        676 = 4x² + 4(x² + 49 + 14x)

⟹        676 = 4x² + 4x² + 196 + 56

⟹        676 = 8x² + 56x+ 196

⟹      8x² + 56x - 480 = 0

⟹           x² + 7x - 60 = 0

⟹       x² + 12x - 5x - 60= 0              [Solving by factorization method]

⟹       x(x + 12) - 5(x + 12) = 0

⟹        (x + 12)(x - 5) = 0

⟹         x = -12 or 5

Ignoring x = -12, as distance cannot be negative, we have:

            x = 5

∴ AC = 2x = 10km

And BC = 2(x + 7) = 2(5 + 7) = 24km

So, distance covered from A to B via C= AC + BC = 10 + 24 = 34km

And, distance covered directly from A to B = AB = 26km

Hence, the distance saved = 34km - 26km = 8km.