# 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 the possible to do so? If yes, at what distances from the two gates should the pole be erected?

By K Balaji|Updated : November 12th, 2022

The pole should be erected at 12 m and 5 m from the two gates.

Let A and B be the opposing fixed gates and P be the pole that needs to be built.

Given that PA – PB = 7

Let PA = a, PB = b

Hence a – b = 7

⇒ a = 7 + b ..(1)

In right triangle PAB

AB2 = AP2 - BP2 (By Pythagoras Theorem)

Substituting the values we get

132 = a2 - b2

169 = (7 + b)2 + b2

169 = 49 + 14b + 2b2

2b2 + 14b - 120 = 0

Taking common from above equation we get

b2 + 7b - 60 = 0

b2 + 12b - 5b - 60 = 0

Taking common we get

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

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

b = 5 or b = -12

As b cannot be negative

Therefore, b = 5, a = 7 + 5 = 12

Hence, PA = 12 m and PB = 5 m

### Pythagoras Theorem Statement

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse. Due to its position opposite the 90° angle, the hypotenuse in this case is the longest side. When the positive integer sides of a right triangle (let's say sides a, b, and c) are squared, the result is an equation known as a Pythagorean triple.

Summary:-

## 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 the possible to do so? If yes, at what distances from the two gates should the pole be erected?

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. The pole should be erected at 12 m and 5 m from the two gates.

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