Question

A circle is inscribed in an equilateral triangle .If the radius of the circle is 5 cm,then what ia the length of the sides of a triangle?​

Answer 1

Answer:

Let the side of the triangle equal x and let the height equal h. Then the area

A=12xh.

Since we are dealing with an equilateral triangle, the bisectors are medians and they are also heights (perpendicular to their bases), and they all intersect in the center of the inscribed circle. Using Pythagorean theorem on half of the big triangle we get

h2+(x2)2=x2.

From where

h2=34x2, i.e. h=3–√2x.

So, the area

A=12xh=A=12x⋅3–√2x=3–√4x2.

Now, to find x we look into the little red triangle. Again using the Pythagorean theorem we have

(h−10)2=(x2)2+102.

Substituting 3√2x for h we get

(3√2x−10)2=x24+100.

Solve for x

3x24−103–√x+100=x24+10012x2–103–√x=0x(12x−103–√)=0x=0 or x=203–√.

Since we are not looking for the trivial solution with x=0, we calculate the area using x=203–√.

So, the area

A=3–√4x2=3–√4⋅400⋅3=3003–√.

hope it's helpful

Mark as Brainliest