A triangle has sides with lengths 7, 24, and 25. 1

A triangle has sides with lengths 7, 24, and 25. 1. Verify this is a Pythagorean triple. 2. Approximate the acute angles in this triangle.

The correct answer and explanation is:

Let’s analyze the triangle with side lengths 7, 24, and 25.

1. Verify if this is a Pythagorean triple

A Pythagorean triple is a set of three positive integers a,b,ca, b, c such that: a2+b2=c2a^2 + b^2 = c^2

where cc is the longest side (hypotenuse).

Given the sides: 7, 24, and 25, the longest side is 25. Let’s check: 72+242=?7^2 + 24^2 = ?

Calculate each: 72=497^2 = 49 242=57624^2 = 576

Sum: 49+576=62549 + 576 = 625

Now calculate 25225^2: 252=62525^2 = 625

Since 72+242=2527^2 + 24^2 = 25^2

this confirms that (7, 24, 25) is a Pythagorean triple, meaning this triangle is a right triangle.

2. Approximate the acute angles in this triangle

Since it is a right triangle, one angle is 90∘90^\circ. The other two angles are acute and sum to 90∘90^\circ.

Let’s find the angles opposite the sides 7 and 24 using trigonometric functions. Let:

• θ\theta be the angle opposite the side of length 7,
• ϕ\phi be the angle opposite the side of length 24.

Using sine or tangent: sin⁡θ=oppositehypotenuse=725=0.28\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{25} = 0.28 θ=sin⁡−1(0.28)≈16.26∘\theta = \sin^{-1}(0.28) \approx 16.26^\circ

For the other acute angle: ϕ=90∘−θ≈90∘−16.26∘=73.74∘\phi = 90^\circ – \theta \approx 90^\circ – 16.26^\circ = 73.74^\circ

Or using tangent: tan⁡θ=724≈0.2917\tan \theta = \frac{7}{24} \approx 0.2917 θ=tan⁡−1(0.2917)≈16.26∘\theta = \tan^{-1}(0.2917) \approx 16.26^\circ

Both methods agree.

Summary

• The triple (7, 24, 25) satisfies the Pythagorean theorem, confirming it is a right triangle.
• The acute angles are approximately 16.3∘16.3^\circ and 73.7∘73.7^\circ.

Explanation

The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. By calculating 72+2427^2 + 24^2 and confirming it equals 25225^2, we prove the triangle is right-angled.

Knowing one angle is 90∘90^\circ, the other two angles must add to 90∘90^\circ. Using trigonometric functions such as sine or tangent with the known sides allows us to find the precise measures of these angles. Calculations show one angle is about 16.3°, making the other 73.7°. This knowledge helps in many practical fields such as construction, navigation, and physics, where accurate angle measurements are essential.