An equilateral triangle and a regular hexagon have the same area. The ratio of the side length of the regular hexagon to the side length of the equilateral triangle can be written as m/√n. What is m+n? Help me plz!

7

Step-by-step explanation:

Let a and b be the sides of the equilateral triangle and the regular hexagon respectively as shown in figures (1) and (2).

The area of the equilateral having side a is

Now, join all the 6 corner points of the regular hexagon to the center of the hexagon as shown in figure (3).

As there are 6 equal sides, so angle subtended by each side at the center is 360/6=60 degrees.

Considering the triangle POR:

degree [angle at center]

As sides PO=RO, so degree.

Hence, triangle POR is an equilateral triangle.

Similarly, all the remaining 5 triangles are equilateral triangles.

So, the area of hexagon = 6 x Area of triangle POR

As the length of sides of the triangle POR is b,

so the area of the triangle POR .

Hence, the area of the regular hexagon,

As the equilateral triangle and a regular hexagon have the same area, so from the equations (i)and (ii), we have

Given that the ratio of the side length of the regular hexagon to the side length of the equilateral triangle, b/a= m/√n.

On comparing with the equation (iii), we have

m=1 and n=6

Hence, m+n=1+6=7.