SOLUTION: a landscape architect designed a flower bed in a shape of a triangle with side lengths 12,14,15.the architech needs to know the area of the flower bed. My Notes to you: it shows

Algebra ->  Triangles -> SOLUTION: a landscape architect designed a flower bed in a shape of a triangle with side lengths 12,14,15.the architech needs to know the area of the flower bed. My Notes to you: it shows      Log On


   

Question 26185: a landscape architect designed a flower bed in a shape of a triangle with side lengths 12,14,15.the architech needs to know the area of the flower bed.
My Notes to you: it shows a triangle. base is 14 and the sides are 13 and 15. with a line down the middle.
I belive Im suppose to use the pangerium therom
All I need to know how to find the height.
Ive tried to split 14 in different ways but the height is not the same on both sides
Thanks for all the help!!!
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Well, you must be studying a new theorem because I've never heard of the "Pangerium theorem"! Perhaps you meant to write "Pythagorean theorem" but if the problem is to find the area of the triangle, you don't need to find the height since you are given the lengths of the three sides. You can use Heron's formula for finding the area if you know the lengths of the three sides. a = 13, b = 14, and c = 15
Heron's formula:
A+=+sqrt%28S%28S-a%29%28S-b%29%28S-c%29%29 Where: S is the semi-perimeter of the triangle and a, b, and c are the lengths of the sides.
S+=+%281%2F2%29%28a%2Bb%2Bc%29
S+=+%281%2F2%29%2813%2B14%2B15%29
S+=+%281%2F2%29%2842%29
S+=+21
A+=+sqrt%2821%2821-13%29%2821-14%29%2821-15%29%29
A+=+sqrt%2821%288%29%287%29%286%29%29

A+=+sqrt%287056%29
A+=+84sq.units.
But now, let's try to find the height of the scalene triangle. The base is 14, so when we drop a perpendicular line from the vertex angle to the base, the base is split into two unequal parts.
Let's label one part x. The other part will then be 14-x. So we now have two right triangles whose dimension are:
1) Hypotenuse = 13, base = x, and the height = h.
2) Hypotenuse = 15, the base = 14-x, and the height = h (same as the h above)
Now we can use the Pythgorean theorem c%5E2+=+a%5E2+%2B+b%5E2to find the height, h.
For triangle 1):
13%5E2+=+h%5E2+%2B+x%5E2 Solve this for h%5E2.
h%5E2+=+13%5E2+-+x%5E2
h%5E2+=+169+-+x%5E2
For triangle 2):
15%5E2+=+h%5E2+%2B+%2814-x%29%5E2 Solve this for h%5E2.
h%5E2+=+15%5E2+-+%2814-x%29%5E2
h%5E2+=+225+-+%28196+-28x+%2B+x%5E2%29 Simplify.
h%5E2+=+29+%2B+28x+-+x%5E2 But h%5E2+=+h%5E2, so...
169+-+x%5E2+=+29+%2B+28x+-+x%5E2 Simplify and solve for x. Add x%5E2 to both sides.
169+=+29+%2B+28x Subtract 29 from both sides.
140+=+28x Divide both sides by 28.
5+=+x or x+=+5 Now substitute this value into one of the two equations for h%5E2 and solve for h.
h%5E2+=+169+-+x%5E2
h%5E2+=+169+-+5%5E2
h%5E2+=+169+-+25
h%5E2+=+144 Take the square root of bothsides.
h+=+12 Discard the negative solution as the height can only be positive.
So, finally, we have the height of the scalene triangle and now we can find the area using: A+=+%281%2F2%29bh where b = 14 and h = 12.
A+=+%281%2F2%29%2814%29%2812%29
A+=+84 sq. units.