Question

A circle is inscribed in a square whose length of the diagonal is 16 cm. An equilateral triangle is inscribed in that circle. What is the length of the side of triangle ?

Answer 1

Step-by-step explanation:

Diagonal of the square is 16 cm

Let side of the square be “a”

We know that each side of a square is equal to one another and that the sides subtend right angles with each other

=> $a^{2} = 16^{2}$

=> a = 4 cm

Side of the square = diameter of the circle

=> diameter = 4 cm

=> radius = 2 cm

Radius of the circle = distance from the centroid to the vertex of an equilateral triangle

=> distance from centroid to vertex = 2 cm

We know that the medians, altitudes and perpendicular bisectors are the same for an equilateral triangle

We also know that the centroid divides the median in the ratio 2:1

Therefore, the altitude of the triangle = height of the triangle = 2+1 = 3 cm

Formula for the height of an equilateral triangle is

$\frac{\sqrt{3}*a }{2}$ where a = side of the equilateral triangle

Substituting the value of the height and solving, we get

=> $a = 2\sqrt{3}$

Therefore the side of the triangle is $2\sqrt{3}$

Please brainlist my answer, if helpful!