.(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...

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.