Find width and length of rectangle given diagonal and area

Question

The diagonal of rectangle is 25, its area is 168, find width and length. I tried solving this problem using trigonometry since diagonal and two sides forms a right triangle, from area i got that a=168/b, considering that area is a*b=168, I reached to a quadratic equation but i got a negative root, does anyone have any idea? Besides the solutions are a=7 and b=24.

Answer 1

We have $$ab=168\tag 1$$ $$a^2+b^2=25^2\tag2$$ So, from $(1)(2)$, we have $$(a+b)^2=a^2+b^2+2ab=25^2+2\cdot 168=961\Rightarrow a+b=31.\tag3$$

Now from $(1)(3)$, we know that $a,b$ are the solutions of $$x^2-31x+168=0,$$ i.e. $$(x-7)(x-24)=0.$$

Answer 2

It normal to find negative solution, they do mathematics. In real world of course lengh is positive !

Just look $$7^2+24^2=25^2=(-7)^2+24^2=7^2+(-24)^2$$

Hope it helps you ;)

Answer 3

You have two equations $$ab = 168 \implies a = \frac{168}{b}$$ and $$a^2 + b^2 = 25^2$$

Substituting the first into the second and multiplying throughout by $b^2$ yields $$\frac{168^2}{b^2} + b^2 = 25^2 \implies 168^2 + b^4 = 25^2b^2$$ This is a quadratic in $b^2$ that gives us solutions $$b^2 = 49 \quad \text{ or } \quad 576$$

Hence $b = \pm 7$ and $b = \pm 24$. We neglect the negative solutions to get $$a = 24, b=7 \quad \text{or} \quad a=7, b= 24$$

Which is just symmetric. So you can simply say that one side is $24$ and the other is $7$.