Question

A triangle has sides with lengths 5, 12, and 13. Verify this is a Pythagorean triple. Approximate the acute angles in this triangle.​

Answer 1

A triangle with lengths 5, 12, and 13 is a Pythagorean triple

The approximate acute angles are 22.6° and 67.4°

Pythagorean triple

Pythagorean triple consist of positive number(a, b, c) such that it obeys the rule:

• c² = a² + b²

A triangle whose sides form a Pythagorean triple is called a right angle triangle.

The longest side is the hypotenuse side.

Therefore,

• 5² + 12²
• 25 + 144 = 169
• √169  = 13

Therefore,

5² + 12² = 13²

This means the triangle is a Pythagorean triple.

Acute angles

Acute angles are angles that are less than 90 degrees. This means the other two angles are acute angle.

Therefore, let's find them

• tan ∅ = opposite / adjacent = tan ∅ = 5 / 12 = ∅ = 22.6198649155 = 22.6°
• 180 - 22.6 - 90 = 67.4°

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Answer 2

The given numbers (can be taken as) the length of the three sides of a

(right) triangle.

Correct response:

• 13² = 5² + 12², therefore,  5, 12, and 13 are a Pythagorean triple

• The acute angles are; θ₁ ≈ 67.38°, and θ₂ ≈ 22.62°

Method of analysis to verify a Pythagorean triple

A Pythagorean triple are three positive integers, a, b, and c that are

related as follows;

c² = a² + b²

Therefore;

$c = \mathbf{\sqrt{a^2 + b^2}}$

Where;

a = 5, b = 12, we have;

$\mathbf{\sqrt{5^2 + 12^2} } = 13 = c$

Therefore;

• The three numbers form a Pythagorean triple, the hypotenuse side (longest side = c = 13)

The acute angles are found as follows;

From trigonometric ratio, we have;

$An \ tangent \ of \ acute \ angle \ \theta_1, \ \tan(\theta_1) = \dfrac{12}{5} = \mathbf{ 2.4}$

Which gives;

• An acute angle of the right triangle, θ₁ = arctan(2.4) ≈ 67.38°

The acute angles of a right triangle are complimentary.

Therefore;

• The other acute angle is therefore, θ₂ ≈ 90° - 67.38° = 22.62°

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