The area of a triangle inscribed in a circle problem

Question

Problem: The area of a triangle inscribed in a circle having a radius $9 cm$ is equal to $43.23 sq. cm$. If one of the sides of the triangle is $18 cm.$, find one of the other side.

To solve the problem, It was assumed that the triangle is a right triangle, and that the given side of the triangle in the problem ($18 cm$) is set as the hypotenuse.

Question: What part of the problem gives a clue that the triangle is a right triangle and that the $18 cm$ side is the hypotenuse of the triangle? Because it could be an isosceles or equilateral or other triangle shapes.

Answer 1

Look, it is said that one side = 18cm which is 2 times the radius. So the triangle you were given was inscribed in a semicircle. Now it is a property of the circle subtended by the diameter of the circle must equal 90 degrees. That is why the triangle is assumed to be a right triangle.

https://en.wikipedia.org/wiki/Thales%27s_theorem

Answer 2

One side of the triangle is $18$ cm, that is one of the side is the diameter. Also note that the longest chord is a diameter.

We know that a triangle inscribed in a circle having a diameter as one side is a right triangle