Introduction to the Problem

Given a triangle with side lengths 24 cm, 30 cm, and 36 cm, this article will guide you in finding the radius of a circle that is tangent to the shortest (24 cm) and longest (36 cm) sides of the triangle. The circle's center lies on the third side (30 cm). This geometric construction involves the application of triangle properties, tangency, and trigonometric identities.

Step-by-Step Solution

Step 1: Basic Information

The given triangle has side lengths:

AB 24 cm AC 30 cm BC 36 cm

Let O be the center of the circle on AC such that the circle is tangent to AB and BC. The circle touches BC and AB at points M and N respectively, with OM ON R (the radius of the circle).

Step 2: Finding the Radius Using Heron's Formula

Heron's formula is used to calculate the area of the triangle. First, calculate the semi-perimeter:

2s 24 30 36 90

The area (A) of the triangle using Heron's formula is:

A 1/4 * sqrt{(90 * (90 - 24) * (90 - 30) * (90 - 36))} 135 * sqrt(7) sq cm

The area can also be expressed as:

A 1/2 * 36 * R 1/2 * 24 * R 30R

Equate and solve for R:

30R 135 * sqrt(7) 135/7 cm2

R 135 / (7 * 30) 9 * sqrt(7) / 2 11.9 cm

Step 3: Using Trigonometric Identities

Note that the radius of the circle is half the height of the triangle. This is due to a specific relationship with the lengths of the sides (2c ab in the general case). Let's explore the trigonometric approach:

Using the tangent and secant identities for angles A and C in the triangle, we can write:

tan(A/2) r/(36 - p)

tan(C) r/p

Using the cosine rule, we get:

cos(A) (242 362 - 302) / (2 * 24 * 36) 9/16

cos(C) (302 362 - 242) / (2 * 30 * 36) 3/4

sec(C) 4/3

Now, equating the expressions for the tangents and secants, we find:

r 9 * sqrt(7) / 2

This step-by-step process ensures a comprehensive understanding of the geometric and trigonometric principles involved in solving such problems.

Conclusion

The radius of the circle tangent to the shorter and longer sides of the triangle with side lengths 24 cm, 30 cm, and 36 cm is r 9 * sqrt(7) / 2. This problem showcases the application of triangle properties, geometry, and trigonometric identities in solving complex geometric constructions.