.(a) In APQR, ∠PQR= 90°. If its area is 216cm\(^2\) and |PQ|:|QR| is 3:4, find |PR|. (b) The present ages of a man and his son are 47 years and 17 years res...

.(a) In APQR, ∠PQR= 90°. If its area is 216cm\(^2\) and |PQ|:|QR| is 3:4, find |PR|.

(b) The present ages of a man and his son are 47 years and 17 years respectively. In how many years would the man's age be twice that of his son?

Answer Details

(a) We can use the formula for the area of a triangle, which is A = 1/2 * base * height, where the base is PQ and the height is PR. Since we know that PQ:QR is 3:4, we can let PQ be 3x and QR be 4x for some value of x.

Then, using the Pythagorean theorem, we can find PR: PR² = PQ² + QR² = (3x)² + (4x)² = 25x².

Substituting this into the formula for the area, we get:

216 = 1/2 * 3x * PR
PR = 144/3x = 48/x

So, we need to find x to determine PR. To do this, we can use the fact that the area of the triangle is also equal to 1/2 * PQ * PR. Substituting in the values we know, we get:

216 = 1/2 * 3x * PR
216 = 3/2 * x * (48/x)
216 = 72
x = 3

Therefore, PR = 48/3 = 16 cm.

(b) Let x be the number of years from now when the man's age will be twice that of his son. Then, in x years, the man's age will be 47 + x and the son's age will be 17 + x.

We can set up an equation using the fact that the man's age will be twice that of his son:

47 + x = 2(17 + x)

Solving for x, we get:

x = 13

Therefore, in 13 years, the man's age will be twice that of his son.