# Math Help - Square

1. ## Square

A B C D is a square

Find the Area of triangle AYX

2. Hello, Singular!

I found a solution, but I hope there is a better way . . .

ABCD is a square.
Find the area of triangle AYX.
Code:
           q    X    s-q
C * - - - - * - - - - * D
|        :::*       |
|       *:::::*  4  | s-p
|  5   :::::::::*   |
|     *:::::::::::* |
s |    :::::::::::::::* Y
|   *:::::::::::*   |
|  :::::::::*       | p
| *:::::*      3    |
|:::*               |
A * - - - - - - - - - * B
s

Let s = side of the square.
. . We see that: . .= .∆ + 12 .[1]

Let YB = p, then DY = s - p
Let CX = q, then XD = s - q

From right triangle YBA: .½ps = 3 . . ps = 6---[2]
From right triangle XCA: .½qs = 5 . . qs = 10 .[3]
From right triangle XDY: .½(s - p)(s - q) = 4 . . s² - ps - qs + pq .= .8 .[4]

Substitute [2] and [3] into [4]: .s² - 6 - 10 + pq .= .8 . . s² + pq .= .24 .[5]

Multiply [2] and [3]: .pqs² = 60 . . pq = 60/s²

Substitute into [5]: .s² + 60/s² .= .24 . . s^4 - 24s² + 60 .= .0
. . . . . . . . . - . . . - . . . - . . . . . .____________
. . . . . . . . . . . . . . . . . . . 24 ± √24² - 4(1)(60) . . . . . . . . .__
Quadratic Formula: . .= .------------------------- .= .12 ± 2√21
. . . . . . . . . . . . . . . . . . . . . . . . 2(1)

. . . . . . . . . . . . - . . . - . . . . . . . .__
Equate to [1]: . ∆ + 12 .= .12 ± 2√21
. . . . . . . . . . . . . . . . .__
. . Therefore: . .= .2√21

3. I've got a doubt, what does mean 3, 4 and 5?, sides of the triangle?

In that case it'd be a right triangle...

4. Originally Posted by Krizalid
I've got a doubt, what does mean 3, 4 and 5?, sides of the triangle?

In that case it'd be a right triangle...
Look at the units in the original post. The units are in cm^2, which designates an area. So I'd say that the area of the triangle is the area of the square (whatever it is) minus 3 cm^2 + 4 cm^2 + 5 cm^2.

Looking at it, Soroban likely has the solution, unless there is some kind of clever shortcut.

-Dan

5. Jejeje

Thanks, first time that I see a problem with that notation.

6. Thanks SOROBAN