Question 163039
Okay, here we go.  Not a difficult problem, but it comes in five parts.  Here we go…
First of all, you must draw your picture correctly according to the description:
First, draw an isosceles triangle, but don’t worry about the dimensions, as long as you determine which side is the base.  Now label it correctly.  You have the base on the bottom, labeled F and G.
The top of the triangle is E.  Now you are given the two midpoints of the sides.  Now, a midpoint divides the side in question into two equal parts, and since this is an isosceles triangle, both of the sides of the triangle (EF and EG) are equal, and thus, since each of two equal sides are cut exactly in half, each of those 4 segments are equal.  Now we can answer the first part:
WHAT IS THE VALUE OF x?
Since each of the four segments are equal, and the lengths of two of them are given (2x+3 and 5x-9), we can set these equal to each other to solve for x:
So, 2x+3 = 5x-9.  Solving for x gives you that x=4.  That is the answer to the first question.
FIND THE PERIMETER OF TRIANGLE EFG.
Since we now know that x=4, then the lengths of each of the side segments is 11, and thus each of the equal sides of the isosceles triangle is 22.  You’re also given that the base is equal to 4x, and since we know that x=4, we know that the base is 16.  Since the perimeter of a triangle (or any figure) is the sum of the sides, then the perimeter of this triangle is 22 + 22 + 16, or 60.
FIND THE AREA OF THE TRIANGLE USING PYTHAGORAS.  ROUND TO THE NEAREST WHOLE NUMBER.
The Pythagorean theorem says that, in a right triangle, the square of the hypoteneuse is equal to the sum of the squares of the other two sides.
Thus, we have to find the height.  The math to prove this is too long to show here, but suffice to say that the line dropped from the vertex of the third angle of an isosceles triangle does a few things.  It bisects that angle, it forms a right angle with the base, and it bisects the base.  Since it bisects the base, and we know that the base = 16, each side of the base on either side of the right angle = 8.  We then now know two of the three sides of the right triangle.  The hypoteneuse is 22, the base is 8, and, thus, we can compute the value of the third side, which is the height of the big triangle.
So, we have that 8^2 + x^2 = 22^2.
This gives us 64 + x^2 = 484.  Subtract 64 from both sides, and this gives us:
x^2 = 420.  Now take the square root of both sides and this gives us that:
x = 20.49.
Now we have the base and the height forming a right angle with the base.  We can then use the formula to find the AREA of the triangle using the formula:
AREA = ˝ (BASE)(HEIGHT):
AREA = ˝ (16)(20.49) = 163.95 = 164 (the answer was asked to be rounded to the whole number).
FIND THE AREA OF THE TRIANGLE USING HERO’S FORMULA
Hero’s formula says that if S = (the sum of the sides of the triangle) divided by 2, then the AREA of the triangle can be given by:
AREA = the square root of [S (S – side one)(S – side 2)(S – side 3)]
So, in this case, S = (22 + 22 + 16)/2, which is 60/2, or 30.
Thus the AREA = the square root of [30(30 – 22)(30 – 22)(30 – 16)], which is sqrt[30(8)(8)(14)], which is sqrt[26880], which is 163.95 = 164 (the answer was asked to be rounded to the whole number).
HOW DO THEY COMPARE?
They are the same.

WHEW !!!  Hope that this  helps.  Good luck !