The length and the breadth of a rectangle are 24cm and (13+r)cm respectively. If the length of the diagonal of the rectangle is (17-3r)cm, find r.

Answer 1

The length and the breadth of a rectangle are 24cm and (13+r)cm respectively. If the length of the diagonal of the rectangle is (17-3r)cm, find r.

Let length = L and given L = 24 cm

Let breadth = B and given B = (13 + r) cm

Let length of the diagonal = D = (17 - 3r)

The diagonal D forms a right triangle with hypotenuse D and legs equal to L and B,

so by the Pythagorean Theorem:

(L)² + (B)² = (D)²

^{Substitute in for L, B, and D:}

^{(24)2 + (13 + r)2 = (17 - 3r)2}

^{576 = 169 + 26r + r2 = 289 + 102r + 9r2}

^{-9r2 - 76r + 287 = 0}

^{Use the quadratic formula here with}

^{a = -9}

^{b = -76}

^{c = 287}

^{Make it easy on yourself, use this website to calculate r quickly rather than using the quadratic formula and doing it with a calculator or by hand:}

https://www.calculatorsoup.com/calculators/algebra/quadratic-formula-calculator.php

^{r = -11,3 or 2.8}

L,W, and D cannot be negative, but it is POSSIBLE that r is. We have to plug in both values for r and check to see which one fits the given data.

breadth = (13 + r)

so if r = -11.3, then breadth = 1.7

and if r = 2.8, then breadth = 15.8

length = 24

so if breadth = 1.7, the length is 24 and diagonal is 17 - 5.1 = 11.9

and if breadth = 15.8 the length is 24 and diagonal is 17 - 47.4 = -30.4

diagonal cannot be negative, so the diagonal = -30.4 is ignored and r = 2.8 for this possibility.

therefore r must = -11.3

r = -11.3 is the answer that you seek.

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Thanks!

John B.

Atlanta, GA

Answer 2

a=13+r

b=24

c=17-3r

a^2+b^2=c^2

(13+r)^2+24^2=(17-3r)^2

(169+26r+r^2)+576=289-102r+9r^2

745+26r+r^2=289-102r+9r^2

456=-128r+8r^2

r=19