Question

if the altitude of an equilateral triangle is 12underoot 3 then what are its area and perimeter​

Answer 1

Answer:

an equilateral triangle, the altitude drawn from a vertex bisects the opposite side.

Proof. Consider an equilateral triangle ABC and D be the foot of the altitude drawn from the vertex A on the opposite side BC. Now, ADB and ADC are right triangles, so we use the Pythagorean theorem to get,

AD¯2+BD¯2=AB¯2andAD¯2+DC¯2=AC¯2

Since ABC is equilateral, we have AB¯=AC¯ , so from above, we get AD¯2+BD¯2=AD¯2+DC¯2 .

Cancelling AD¯2 from both sides, we get BD¯2=DC¯2 . Since sides lengths are non-negative (positive except in cases of degeneracy), taking square roots give BD=DC which shows D, the foot of the altitude drawn from A bisects the opposite side BC. A similar argument works for the other two vertices as well.

QED

Now that we’ve established this, the conclusion is the same as the answer by Esha: We apply the Pythagorean theorem to the triangle ADB where AD=sqrt(3), AB=k and DB=k/2; we get 3+k2/4=k2⟹k2=4⟹k=2⟹Perimeter=3k=6