Discovering Similar Triangles: A Right-Angled Triangle's Lengths

Understanding the principles of similar triangles and the Pythagorean theorem can help solve a variety of geometrical problems. In this article, we'll walk through the process of finding the lengths of the legs of a similar right-angled triangle when given the lengths of the original triangle's legs and the hypotenuse of the similar triangle.

Introduction to Right-Angled Triangles and the Pythagorean Theorem

A right-angled triangle is a triangle with one angle measuring 90°. The longest side of this triangle, which is opposite the right angle, is known as the hypotenuse. The Pythagorean theorem, named after the Greek mathematician Pythagoras, defines the relationship between the lengths of the sides of a right-angled triangle. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (legs).

Calculating the Original Triangle's Hypotenuse

To solve the problem of finding the lengths of the legs of a similar right-angled triangle, we start by calculating the hypotenuse of the original right-angled triangle. Given the lengths of the legs as 24 cm and 10 cm, we use the Pythagorean theorem:

Step 1: Calculate the Hypotenuse

Using the formula for the Pythagorean theorem, we calculate the hypotenuse (c) of the original right-angled triangle:

c √(a2 b2)

c √(242 102)

c √(576 100) √676 26 cm

Thus, the hypotenuse of the original right-angled triangle is 26 cm.

Identifying Scaling Factors in Similar Triangles

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. This means that the ratios of the lengths of the corresponding sides of the two triangles are equal.

Step 2: Determine the Scaling Factor

Given that the hypotenuse of the similar triangle is 52 cm, we can determine the scaling factor (k) by using the ratio of the hypotenuse of the similar triangle to the hypotenuse of the original triangle:

Hypotenuse Scaling Factor

k (Hypotenuse of similar triangle) / (Hypotenuse of original triangle) 52 cm / 26 cm 2

Therefore, the scaling factor (k) is 2.

Finding the Legs of the Similar Triangle

With the scaling factor known, we can now determine the lengths of the legs of the similar triangle. The lengths of the legs of the similar triangle are obtained by multiplying the lengths of the corresponding legs of the original triangle by the scaling factor.

Step 3: Calculate the Legs of the Similar Triangle

Leg of similar triangle 24 cm × k 24 cm × 2 48 cm

Leg of similar triangle 10 cm × k 10 cm × 2 20 cm

Therefore, the lengths of the legs of the similar triangle are 48 cm and 20 cm.

Conclusion

By applying the principles of the Pythagorean theorem and understanding the concept of similar triangles, we can easily determine the lengths of the legs of a similar right-angled triangle. This problem provides a practical example illustrating the real-world applications of these mathematical concepts.