Finding Inradius Of Similar Triangles A Step-by-Step Solution
In geometry, understanding the properties of similar triangles is crucial for solving various problems related to areas, perimeters, and inradii. This article delves into a specific problem involving two similar triangles, ABC and DEF, where their areas and the difference in their inradii are given. We aim to find the inradius of triangle ABC. This detailed explanation will not only provide a step-by-step solution but also enhance your understanding of the underlying geometric principles.
Problem Statement
Triangle ABC is similar to triangle DEF. The area of triangle ABC is 144 cm², and the area of triangle DEF is 225 cm². The inradius of triangle DEF is 2.5 cm more than the inradius of triangle ABC. Find the inradius of triangle ABC.
Understanding Similar Triangles
Before diving into the solution, it's essential to grasp the concept of similar triangles. Two triangles are said to be similar if their corresponding angles are equal, and their corresponding sides are in proportion. This proportionality extends to various other properties, including altitudes, medians, perimeters, and inradii. Crucially, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides or any other corresponding linear dimensions, such as inradii. This principle forms the backbone of our solution.
Key Concepts and Formulas
To solve this problem effectively, we need to be familiar with the following concepts and formulas:
- Similarity of Triangles: If triangle ABC is similar to triangle DEF (denoted as ΔABC ~ ΔDEF), then the ratios of their corresponding sides are equal. For instance, AB/DE = BC/EF = CA/FD.
- Ratio of Areas of Similar Triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Mathematically, Area(ΔABC) / Area(ΔDEF) = (AB/DE)² = (BC/EF)² = (CA/FD)².
- Inradius of a Triangle: The inradius (r) of a triangle is the radius of the circle inscribed within the triangle. The area of a triangle can be expressed in terms of its inradius and semi-perimeter (s) as Area = r * s, where s = (a + b + c) / 2, and a, b, and c are the sides of the triangle.
- Ratio of Inradii of Similar Triangles: The ratio of the inradii of two similar triangles is equal to the ratio of their corresponding sides. If r1 and r2 are the inradii of ΔABC and ΔDEF respectively, then r1/r2 = AB/DE = BC/EF = CA/FD.
Step-by-Step Solution
Now, let's solve the problem step by step, applying the concepts and formulas mentioned above.
Step 1: Determine the Ratio of Areas
We are given that the area of triangle ABC is 144 cm² and the area of triangle DEF is 225 cm². The ratio of their areas is:
Area(ΔABC) / Area(ΔDEF) = 144 / 225
Step 2: Simplify the Ratio of Areas
Simplify the fraction 144/225 by finding the greatest common divisor (GCD) of 144 and 225. The GCD is 9. Divide both the numerator and the denominator by 9:
144 / 225 = (144 ÷ 9) / (225 ÷ 9) = 16 / 25
Step 3: Find the Ratio of Corresponding Sides (or Inradii)
Since the ratio of the areas of similar triangles is the square of the ratio of their corresponding sides (or inradii), we can write:
(r1 / r2)² = 16 / 25
Taking the square root of both sides, we get:
r1 / r2 = √(16 / 25) = 4 / 5
Here, r1 is the inradius of triangle ABC, and r2 is the inradius of triangle DEF.
Step 4: Express r2 in Terms of r1
We are given that the inradius of triangle DEF (r2) is 2.5 cm more than the inradius of triangle ABC (r1). So, we can write:
r2 = r1 + 2.5
Step 5: Substitute and Solve for r1
From Step 3, we have r1 / r2 = 4 / 5. We can rewrite this as:
r2 = (5 / 4) * r1
Now, we have two expressions for r2. Equate them:
r1 + 2.5 = (5 / 4) * r1
To solve for r1, first multiply both sides by 4 to eliminate the fraction:
4 * (r1 + 2.5) = 4 * (5 / 4) * r1
4r1 + 10 = 5r1
Now, subtract 4r1 from both sides:
10 = 5r1 - 4r1
10 = r1
So, the inradius of triangle ABC (r1) is 10 cm.
Step 6: Verify the Solution
Let's verify our solution. We found r1 = 10 cm. Then:
r2 = r1 + 2.5 = 10 + 2.5 = 12.5 cm
Now, check if the ratio r1 / r2 is indeed 4 / 5:
r1 / r2 = 10 / 12.5 = 100 / 125 = 4 / 5
Our solution is consistent with the given information.
Conclusion
The inradius of triangle ABC is 10 cm. This problem illustrates how understanding the properties of similar triangles and their relationships can help solve geometric problems. By applying the concepts of ratios of areas and inradii, we were able to find the inradius of triangle ABC effectively. Remember, the key to solving such problems lies in a clear understanding of the fundamental geometric principles and the ability to apply them systematically. This detailed solution not only answers the question but also serves as a valuable guide for tackling similar problems in the future.
Find the inradius of triangle ABC, given that triangle ABC is similar to triangle DEF, the area of triangle ABC is 144 cm², the area of triangle DEF is 225 cm², and the inradius of triangle DEF is 2.5 cm more than the inradius of triangle ABC.
Finding Inradius of Similar Triangles A Step-by-Step Solution