Question**: A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Is it a right triangle? - Coaching Toolbox
Is a Triangle with Sides 7 cm, 24 cm, and 25 cm a Right Triangle? A Clear Geometry Check
Is a Triangle with Sides 7 cm, 24 cm, and 25 cm a Right Triangle? A Clear Geometry Check
When studying triangles, one common question students face is whether a three-sided figure with given side lengths — such as 7 cm, 24 cm, and 25 cm — is a right triangle. In this article, we’ll clearly explain how to determine if this triangle satisfies the key condition of being a right triangle and verify its classification using the Pythagorean theorem.
What is a Right Triangle?
Understanding the Context
A right triangle is defined as a triangle that contains one angle measuring exactly 90 degrees. The side opposite this right angle is called the hypotenuse, the longest side of the triangle. For a triangle with sides a, b, and c (where c is the longest side), it is a right triangle if:
\[
a^2 + b^2 = c^2
\]
This is the well-known Pythagorean Theorem.
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Key Insights
Analyzing the Given Triangle
We are given side lengths: 7 cm, 24 cm, and 25 cm.
First, identify the longest side:
- 7 cm < 24 cm < 25 cm ⇒ The longest side is 25 cm, which we assume could be the hypotenuse.
Now, apply the Pythagorean theorem to test:
\[
7^2 + 24^2 = ?
\]
\[
49 + 576 = 625
\]
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\[
25^2 = 625
\]
Since both expressions are equal:
\[
7^2 + 24^2 = 25^2
\]
This confirms the triangle satisfies the Pythagorean Theorem.
Conclusion: Yes, It Is a Right Triangle
Because the squares of the two shorter sides (7 cm and 24 cm) add up exactly to the square of the longest side (25 cm), this triangle is a right triangle. It has a right angle opposite the 25 cm side, making it a classic example of a 7-24-25 right triangle — one of the well-known Pythagorean triples.
Why This Matters
Understanding whether triangles like this are right-angled is fundamental in geometry, trigonometry, architecture, engineering, and math education. Recognizing right triangles helps in calculating area, verifying structural stability, and solving real-world problems involving angles and distances.
