Find the width and length of a rectangle given only the diagonal
Is it possible to find the width and length of a rectangle by just know the diagonal? Like if the diagonal was 25, what would be the side lengths?
Any help is appreciated!
EDIT: From the comments and solutions this is not possible. Thanks for the explanations!
$\endgroup$ 34 Answers
$\begingroup$Say your diagonal was length $c$, and it’s width and height was $a$ and $b$ respectively, then these must satisfy
$$ a^2 + b^2 = c^2. $$
This has many solutions for $a$ and $b$ given the diagonal $c$.
$\endgroup$ 1 $\begingroup$No. By Pythagoras, if the width is $a$ and the length $b$, then,
$$a^2 + b^2 = 25^2$$
Any pair of numbers that satisfies that equation is a possible rectangle for that diagonal. For example, $a=15, b=20$, or $a=125/13, b=300/13$, etc.
$\endgroup$ $\begingroup$It is not enough to just know the length of the diagonal. But there are different additional hints that can make it possible.
- if the rectangle is a square the side $a=d/\sqrt2$
- if the ratio between the sides $r=a/b$ is known $b=\sqrt{\frac{d^2}{r^2+1}}$
- if the angle $\alpha$ between the diagonals is known $a=d\cdot sin(\frac{\alpha}{2})$
Those are some but not all possible hints that can help.
$\endgroup$ $\begingroup$Another approach. If you know the side of a triangle and the opposite angle, the possible locations of the vertex which isn't on the original side lie on an arc of a circle of known radius. With a right-angle the side you have is a diameter of the circle. The diagonal splits the rectangle into two triangles.
This is a lesser known consequence of the sine rule $$\frac a{\sin A}=2R$$ where $R$ is the radius of the circumcircle of the triangle.
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