MHB Calculate the area of a triangle knowing its perimeter and 2 heights

[loquetedigo]

Calculate the area of a triangle knowing its perimeter and 2 heights

perimeter = 30 m
ha = 8 m
hb = 9 mNOTE = You can use the online triangle calculator TrianCal to see and draw the results.
NOTE = Do not use the values ??of responses.

A) 41.29 m2
B) 42.93 m2 or 36.28 m2
C) 42.95 m2 or 36.29 m2
D) Imposible

 

[Furlo]

Hello, here is what I did (this was the first idea that came to mind and it's kind of a brute force method):

Assuming the sides of the triangle are a,b,c, we have the following equations:

a+b+c+=30

The area of the triangle can be written in 3 different ways.

2 of them using A = Base * Height / 2

A = 8a/2

A = 9b/2

3rd one using Heron's Formula:

Semi-perimeter is 30/2 = 15 so

A = \sqrt{15(15-a)(15-b)(15-c)}

Now, to get an equation in only one unknown, let's pick a.

From equalizing the first 2 formulas for the area, we should get

b = 8a/9

From the perimeter, substituting b and solving for c we should get

c=(270-17a)/9

You should be able to get these answers fairly easy yourself.

Lastly, to write the equation, we equalize the Area that contains a (1st one) and the one from Heron's formula, where we substitute b and c with the values found previously. After simplifying, we should get something like this:

36a=\sqrt{15(15-a)(135-8a)(17a-135)}.

I did't actually bother with trying to solve the equation, but I did graph it using Desmos and found 3 solutions for a
9.071
10.732
20.649

Ignoring the last one (because c would be negative) and substituting both values in the Area formula, after rounding the final answer should be B.