If two of the sides of a right-triangle are 10 cm and 10.5 cm and its inradius is 3 cm, what is its circumradius?

To solve this problem, we need to use the properties of a right triangle and the formulas for the inradius (r) and circumradius (R) of a triangle.

Step 1: Identify the sides of the triangle Given two sides of the triangle are 10 cm and 10.5 cm. Since this is a right triangle, one of these sides is the base (b), the other is the height (h), and the hypotenuse (c) is what we need to find.

Step 2: Find the hypotenuse We can use the Pythagorean theorem to find the hypotenuse: c = sqrt(b² + h²) c = sqrt((10 cm)² + (10.5 cm)²) = 14.5 cm

Step 3: Find the area of the triangle The area (A) of a triangle is given by the formula: A = 1/2 * b * h A = 1/2 * 10 cm * 10.5 cm = 52.5 cm²

Step 4: Find the semi-perimeter of the triangle The semi-perimeter (s) of a triangle is given by the formula: s = (b + h + c) / 2 s = (10 cm + 10.5 cm + 14.5 cm) / 2 = 17.5 cm

Step 5: Use the inradius to confirm the area of the triangle The area of a triangle can also be found by the formula: A = r * s 52.5 cm² = 3 cm * s s = 52.5 cm² / 3 cm = 17.5 cm This confirms our calculation for the semi-perimeter.

Step 6: Find the circumradius The circumradius (R) of a right triangle is given by the formula: R = c / 2 R = 14.5 cm / 2 = 7.25 cm

So, the circumradius of the triangle is 7.25 cm.