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a triangle is show with side lengths 3, 5 and 8. The length 8 is bisected and an unknown length x connects this midpoint and the vertex of the 2 other sides. The solution contends the unknown length is 1.
How can this be? Can you have a triangle if the total length of 2 sides is NOT longer than the 3rd? - No.
The solution must be x=0. intuition also says it must be zero.
Let's use algebra to solve. Construct a line from the vertex of lines 5 &3 perpendicular to the side length 8. call this unknown length b. The distance between the mid-point of the length 8 and the perpendicular line is unknown. call this length a.
using pythagoras. 1. b^2+(4-a)^2=3^2 & 2. b^2+(4+a)^2=5^2.
substitute b^2 from 1 into 2. Solve for 1. a=1
substitute a=1 and equations 1 & 2 give you b = 0. If b=0 then x=0
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