I was solving an exercise on Isosceles trapezoid whose diagonal was given, and I note that If I draw a diagonal in the isosceles trapezoid I got two triangles
To determine the area of the triangles I draw their heights, which are perpendicular to the diagonal. The problem arises when I suppose that the sum of the height of the two triangles is equal to the diagonal. Funny thing I got the correct anwser doing that way, however it was only intuitive, not accurate.
But how to prove if $(a = b + c)$ is true or false ?
Best Answer
Your claim: $a=b+c$ is not true. Since your trapezoid is isosceles, the two lower angles are the same. Therefore, the two diagonals are equal because the lower triangle each creates when dividing the trapezoid are congruent (by SAS). Then, by the Pythagorean theorem, as long as the two perpendiculars are not collinear, $a>b+c$:
Only when the two perpendiculars are collinear will $a=b+c$,
but then your isosceles trapezoid will be a square.(As noted by Issac, the two perpendiculars can be collinear with out the isosceles trapezoid being a square.)