Question

In the figure below, XY ∥ BC. The ratio of the perimeter of triangle ABC to the perimeter of triangle AXY is 3:1. Given that the numerical value of the area of triangle AXY is a whole number, which of the following could be the area of the triangle ABC?
Answer as fast as possible​

Answer 1

Given:

XY || BC

Ratio of perimeter of triangle ABC to perimeter of triangle AXY = 3:1

To find:

Area of triangle ABC

Solution:

Since XY || BC, ΔABC ≈ ΔAXY

Since, the two triangles are similar, then the corresponding sides are equal.

AB = AX

BC = XY

AC = AY

Perimeter of ΔABC = AB + BC + AC

Perimeter of ΔAXY = AX + XY + AY

$\frac{p(ABC)}{p(AXY)}=\frac{3}{1}$     (given)

$\frac{AB+BC+AC}{AX+XY+AY}=\frac{3}{1}$

Let the scale factor be a:b = 3:1

If two triangles are similar and have a scale factor of a:b, then ratio of their areas is a²: b²

$\frac{a(ABC)}{a(AXY)}=\frac{3^{2} }{1^{2} } \\a(ABC)=3^{2}*a(AXY)\\ a(ABC)=9*a(AXY)$

Let a(AXY) be a whole number 'n'

Then, a(ABC) = 9n

Final Answer:

Area of ΔABC is 9n where n is a whole number that gives the area of ΔAXY.