# What is the area of a square drawn inside a triangle?

1. Dec 5, 2015

### diredragon

1. The problem statement, all variables and given/known data
Given the isosceles triangle whose sides are :
c=10
b=13
find the are of a square drawn inside the triangle whose upper edges touch the b sides of the triangle.
2. Relevant equations
3. The attempt at a solution

I named the side of the square a.
First i made two equations that both involve a quantity a. The length of a triangle side b i divided into d and 13 - d. It is divided by the point at which the square touches the side.
$d^2 = a^2 + \frac{(10 - a)^2}{4})$
$(13 - d)^2 = \frac{a^2}{4} + (12 - a)^2$
I cant now figure out how to get a from this. I mean there has to be an easier way than replacing a by d from one of the two expressions. I missed something.

Last edited by a moderator: Dec 5, 2015
2. Dec 5, 2015

### QuantumQuest

I think that it would be more appropriate to draw it and show where is each part. That would be more helpful to you and to anyone want to help, as it will clear out any misunderstandings.

3. Dec 5, 2015

4. Dec 5, 2015

### Samy_A

The small right triangle in the lower left of the drawing is similar to the larger triangle to the left. That should give you 2 linear equations with a and d.

5. Dec 5, 2015

### QuantumQuest

There are many ways to think about this but I think the easiest is what Samy_A suggests.

6. Dec 5, 2015

### diredragon

i get 60/11 to be (a) using the similar triangles.

7. Dec 5, 2015

### Samy_A

I also got that for $a$.

8. Dec 6, 2015

### epenguin

You can maybe simplify calculations etc. by realising that the triangle that is half the given one, with sides 5, 13 has Pythagorean numbers, the other side (height) is 12.

9. Dec 6, 2015

### diredragon

I have already taken that into account and set up a similar triangle property $\frac{13}{d}=\frac{12}{a}$