Finding percentage increase of area

Question

Find the percentage increase in the area of a triangle if its each side is doubled?since no information is given about the type of triangle in question, so should i take a equilateral or isosceles or a scalene triangle, also will all the situations yields the same result? Any suggestion will be helpful.

Answer 1

We have $$A_1=\frac{1}{2}ab\sin(\gamma)$$ then $$A_2=4\frac{1}{2}ab\sin(\gamma)=4A_1$$ Can you finish?

Answer 2

Set b to be the base of the triangle where the other point on the triangle is over b, Then set line h perpendicular to b. Finally set the two angles adjacent to b to π and θ. If you break b up into b1 and b2 (which add up to b) then sin^-1(π)*h=b1 and sin^-1(θ)*h=b2 so (sin^-1(π)*h+sin^-1(θ))*h=b.If the angles π and θ stay the same the h is proportional to b so if b is doubled then h will be doubled too so that means if bh/2 is the original triangle then 2b*2h/2 is the new triangle so we get the new triangle to be 400% the size of the original triangle.

Answer 3

Think about it geometrically. Here we have a triangle:

The type of triangle and its orientation don’t matter since we can perform the following steps starting with any triangle at all.

Now reflect it in the midpoint of $\overline{AB}$, getting a paralellogram:

You will agree, I hope, that this figure has twice the area of the original triangle. Now, chop off and move a right-triangular piece to create a rectangle with the same area:

(All we’ve really done here is to recreate geometrically the formula $\frac12bh$ for the area of a triangle, taking $\overline{AC}$ as the base.)

Now, what happens to the area of this rectangle if you double the lengths of all of those lines? What does this mean for the area of the original triangle?