How do I calculate the surface are of a trapezoid If that trapezoid is inside of a triangle?

[Ana.stasia]

The questiojn is:
Sides of the triangle are 13cm, 14cm and 15cm. The line parallel to the longest side is dividing the triangle so it forms a trapezoid whose sum of all sides in 39cm. Calculate the surface area of the trapezoid.

I did this.
Just in case of a language barrier. P is surface area, O is the sum of all sides, h is the height. (ha is the height for the base side)


I calculated the surface area of the entire triangle and used it to find the height that is used with the base side. However I do not know the rest. Can you guys help me with it?

Thank you in advance

 

[Dr.Peterson]

Suppose the height of the trapezoid is h (unknown). Can you express a+b+c as a function of h? (Use proportions.)

 

[Ana.stasia]

I tried to do this. I am not sure If I got anything useful or how to use it further.

 

[Dr.Peterson]

You haven't yet related c to h_t directly. Use the triangle above c.

Your work is quite different from mine, but equivalent as far as you went.

 

[Ana.stasia]

Thank you so much. I managed to get the correct solution.

 

[Dr.Peterson]

Looks good.

Just to show a different approach, what I did was to find that [MATH]a = \frac{13h_t}{11.2}[/MATH], [MATH]b = \frac{14h_t}{11.2}[/MATH], and [MATH]c = \frac{15(11.2-h_t)}{11.2}[/MATH], and put those into the equation [MATH]a+b+c = 24[/MATH]. Then I could solve for [MATH]h_t[/MATH] and find [MATH]a[/MATH], [MATH]b[/MATH], and [MATH]c[/MATH] from that.