Class 10 RD Sharma Solutions - Chapter 8 Quadratic Equations - Exercise 8.10

Question 1. The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.

Solution:

Let the length of other two sides of the triangle be x and y.

Therefore, according to the question,

x - y = 5

or,

x = y + 5

Now the two sides are of length y and y + 5.

Applying Pythagoras theorem on the right-angled triangle, we get,

25² = y² + (y + 5)²

625 = y² + y² + 25 + 10y

2y² + 25 + 10y = 625

2y² + 10y = 625 - 25

2y² + 10y - 600 = 0

y² + 5y - 300 = 0

y² + 20y - 15y - 300 = 0 [By middle term splitting]

y(y + 20) - 15(y + 20) = 0

(y - 15)(y + 20) = 0

y -15 = 0 or y + 20 = 0

y = 15 or y = -20

As the length of a side is positive, we will take y = 15.

Therefore,

x = y + 5

x = 15 + 5

x = 20

The two sides of the triangle are 15 cm and 20 cm.

Question 2. The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.

Solution:

Let the shorter side of the rectangle be x.

Now, according to the question, length of rectangle is x + 60 and length of the longer side of rectangle is x + 30.

Now, we will apply Pythagoras theorem on the right-angled triangle formed by the diagonal and the two sides,

(x + 60)² = (x + 30)² + x²

x² + 3600 + 120x = x² + 900 + 60x + x²

3600 - 900  = x² + 60x - 120x

2700 = x² - 60x

x² - 60x - 2700 = 0

x² - 90x + 30x - 2700 = 0

x(x - 90) + 30(x - 90) = 0

(x + 30)(x - 90) = 0

x + 30 = 0 or x - 90 = 0

x = -30 or x = 90

We will take positive side, i.e. x = 90

Now, 

length of shorter side is 90 m

length of diagonal is 90 + 60 = 150 m

length of longer side is 90 + 30 = 120 m

Question 3. The hypotenuse of a right triangle is 3√10 cm. If the smaller leg is tripled and the longer leg doubled, new hypotenuse will be 9√5 cm. How long are the legs of the triangle?

Solution:

Let the smaller and longer side of the triangle be x and y.

Now applying Pythagoras theorem, we get,

(3√10)² = x² + y²

x² + y² = 90     ......... (i)

Now the smaller and longer sides are tripled and doubled respectively,

Therefore, the new sides of the triangle will be 3x and 2y.

Applying Pythagoras theorem,

(9√5)² = (3x)² + (2y)²

9x² + 4y² = 405     ......... (ii)

Now multiplying (i) by 4 and then subtracting (i) from (ii), we get

9x² - 5x² + 4y² - 4y²= 405 - 360

5x² = 45

x² = 9

x = 3

Substituting value of x in (i),

9 + y² = 90

y² = 81

y = 9

Length of smaller side is 3 cm.

Length of longer side is 9 cm.

Question 4. A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

Solution:

In the circle, let P be the pole on the circumference of the circle and points A and B be the two diametrically opposite fixed gates.

Let length of PB be x.

Given,

PA - PB = 7

PA = x + PB

PA = x + 7

The triangle PQR is a right-angled triangle at P, as AB is the diameter of the circle.

Now applying Pythagoras theorem,

AB2 = PA2 + PB2

13² = (x + 7)² + x²

169 = x² + 49 + 14x + x²

169 = 2x2 + 14x + 49

2x² + 14x + 49 - 169 = 0

2x² + 14x - 120 = 0

x² + 7x - 60 = 0

x² + 12x - 5x - 60 = 0

x(x + 12) - 5(x + 12) = 0

(x - 5)(x + 12) = 0

x = 5 or x = -12

Only x = 5 is possible.

PB = 5 m

PA = 5 + 7 = 12 m

P should be erected at a distance of 5 m from PB and 12 m from PA.