Calculate area of isosceles Triangle from area of a Square

Question

Please click here to see the Formula Picture

I need to calculate S2 (area of isosceles Triangle) from the formula inside picture . is there any solution to find S2 from informations inside this picture?

thanks

Answer 1

The area of (any) triangle is given by

$$A_{Tri}=\frac{1}{2}bh$$

where $b$ is the length of the base and $h$ is the height. So since $b=a$ and $h=10$, we know that $S_2=\frac{1}{2}10a=5a$.

The area of a square is given by

$$A_{Squ}=s^2$$

where $s$ is the length of the square's sides. So $S_1=a^2$.

Now, plugging in your formulas for $S_i$,

$$S_2=\frac{2}{3}S_1-\frac{8}{3}$$

$$5a=\frac{2}{3}a^2-\frac{8}{3}$$

Now solve for $a$... check both solutions to see if there is a false solution...

EDIT:

For the OP, confusion arose as to how to solve for $a$. Note that

$$5a=\frac{2}{3}a^2-\frac{8}{3}$$

is equivalent to

$$2a^2-15a-8=0$$

which can be solved by factoring or the quadratic formula ($a=2,b=-15,c=-8$). By factoring we see that

$$(2a+1)(a-8)=0 \qquad\Rightarrow\qquad a=-\frac{1}{2}, 8$$

Which one fails?