How do you find the dimensions of a right triangle if a right triangle has area 960 and hypotenuse length = 52?

Question

How do you find the dimensions of a right triangle if a right triangle has area 960 and hypotenuse length = 52?

1 Answer

Impact of this question

3205 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-11T10:30:50

When we solve simultaneous equations

$\frac{1}{2} a b = 960$ $1/2 a b = 960$

$a^{2} + b^{2} = 52^{2}$ $a^2 + b^2 = 52 ^2$

we find no real solutions, so no such right triangle exists.

Explanation:

Call the legs $a$ $a$ and $b$ $b$ .

$\frac{1}{2} a b = 960$ $1/2 a b = 960$

$a^{2} + b^{2} = 52^{2}$ $a^2 + b^2 = 52 ^2$

Let's eliminate $b$ $b$ first.

$b = \frac{1920}{a}$ $b = 1920/a$

$a^{2} + {(\frac{1920}{a})}^{2} = 52^{2}$ $a^2 + (1920/a)^2 = 52^2$

$a^{4} - 52^{2} a^{2} + 1920^{2} = 0$ $a^4 - 52^2 a^2 + 1920^2 = 0$

That's a quadratic equation in $a^{2}$ $a^2$ with a pretty negative discriminant of

$52^{4} - 4 {(1920)}^{2} = - 7433984$ $52^4 - 4(1920)^2 = -7433984$

so no real solutions for $a^{2}$ $a^2$ and thus no real solutions for $a, b$ $a,b$ .

That's the end, but we can look into it a bit deeper.

The hypotenuse is just too small to support this area. Let's find the general formula for the minimum hypotenuse for a real triangle of a given area $A$ $A$ .

$\frac{1}{2} a b = A$ $1/2 ab = A$

$b = \frac{2 A}{a}$ $b = {2A}/a$

$a^{2} + b^{2} = c^{2}$ $a^2 + b^2 = c^2$

$c^{2} = a^{2} + \frac{4 A^{2}}{a^{2}}$ $c^2 = a^2 + {4A^2}/a^2$

$a^{4} - c^{2} a^{2} + 4 A^{2} = 0$ $a^4 - c^2 a^2 + 4A^2 = 0$

The discriminant must be positive or zero:

$c^{4} - 16 A^{2} \geq 0$ $c^4 - 16 A^2 ge 0$

$c^{4} \geq 16 A^{2}$ $c^4 ge 16 A^2$

Everything is positive, so

$c \geq 2 \sqrt{A}$ $c ge 2 sqrt{A}$

In our case

$c = 2 \sqrt{960} = 16 \sqrt{15} \approx 61.97$ $c = 2 sqrt{960} = 16 sqrt{15} approx 61.97$ is the minimum hypotenuse than can support this area, presumably corresponding to the isosceles right triangle with $a = b .$ $a=b.$

Science

Math

Humanities

... and beyond

How do you find the dimensions of a right triangle if a right triangle has area 960 and hypotenuse length = 52?

1 Answer

Explanation:

Related questions

Impact of this question