Appolonius Theorem
You are given a triangle with 2 sides measuring 5 units and 10 units. The median on the third side measures 6.5 units.
If the area of the triangle can be expressed as a x unit 2 where x is square-free, find x .
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4 solutions
Apollonius' theorem
Herons formula?
we can use the steward's theorem(well, apollonius theorem is derived from it)
( Note: because 6.5 is the median of the third side, we can assume the third side has a length of 13 )
By using Heron's formula,
p = 2 a + b + c
A = p ( p − a ) ( p − b ) ( p − c ) , we can solve this problem.
To solve for p:
1 4 = 2 5 + 1 0 + 1 3
To solve for area:
5 0 4 = 1 4 ( 1 4 − 5 ) ( 1 4 − 1 0 ) ( 1 4 − 1 3 )
By simplifying...
6 1 4
STEP 1:Use Appolonius Theorem to find 3rd side of the triangle
STEP 2:Use Heron's Formula for area to find area of the triangle.
It is wrongly worded. The area should be a * sqrt(x).
Extend the median to 6.5 more. Join the end of this line with one of the base vertex. The triangle thus formed will have area equal to given triangle.
The sides are 13, 5, 10...> area = sqrt(14 * 1 * 9 * 4)
= 6 * sqrt(14) = a * sqrt(x) ......> x = 14.
m^2 = (2a^2 + 2b^2 - c^2) /4 where c is the unknown third side.
a= 5, b = 10 and m = 6.5
solving c = 9
using Herons formula, area is sqrt 504 = 6 sqrt 14