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.
[Math] Find width and length of rectangle given diagonal and area
rectangles
Related Solutions
If you are only given one set of sides and the two points, and don't want to say it is square (which would give you the other set of sides), you don't have enough information to solve the problem. The rectangle could be as you have drawn the left picture, or $L1,L2$ (which are equal) could be shorter and the rectangle more tilted.
If you are given both sides as on the right, you have enough information. Do you need the coordinates of the other corners (which are not the $(X1,Y2)$ you label) or is that just a means to get the angle of the bottom side? You can evaluate the angle of he diagonal $(X1,Y1)$ to $(X2,Y2)$ as Atan2(Y1-Y2,X1-X2). Note that this uses the usual math convention of $+x$ to the right, $+y$ up-you may need to reorient it for your coordinates. The angle from the diagonal to the lower side is $\arctan(\frac {70}{60})$ and the angle of the lower side is the sum of these.
From the lower left-hand corner of the rectangle (let's call that point $A$), consider the diagonal line as a secant line of the left-hand circle, intersecting the circle at points $B$ and $C$, where $B$ is between $A$ and $C$. Also consider one of the edges of the rectangle adjacent to $A$ as a tangent line touching the circle at $D$.
Then by a theorem about tangent and secant lines from a point outside a circle, we have the following relationship of the lengths of the segments from $A$ to each of the three points $B$, $C$, and $D$: $$ (AD)^2 = AB \times AC. \tag1 $$
It is easy to find that $AD = \frac12a$. Now let $E$ be the midpoint of the bottom side of the rectangle; then $AC$ is the hypotenuse of right triangle $\triangle AEC$, which has legs $a$ and $\frac12a$, and therefore $AC = (\frac12\sqrt5)a$.
We can then use Equation $(1)$ to find the length $AB$, so we can find the length of the chord $BC$; from that chord and the radius of the circle we can get the angle of $S_1$ at the center of the circle.
Best Answer
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 $7$.