Is 5, 7, and 8 a Right Triangle?

A right triangle is a triangle in which one angle is 90 degrees. The other two angles are acute and the sides that form the right angle are known as the legs, while the longest side is the hypotenuse. To determine if a triangle with sides 5, 7, and 8 is a right triangle, we need to check if it satisfies the Pythagorean theorem. Let's explore this further with the provided information and deepen our understanding of triangle types.

Checking for a Right Triangle Using the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. If this condition is not met, the triangle is not a right triangle.

Let's consider the sides 5, 7, and 8:

t52 72 25 49 74 t82 64 t74

Since 74 is not equal to 64, the sides 5, 7, and 8 do not form a right triangle. Let's explore this in more detail:

Is 5, 7, and 8 a Right Triangle?:

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Consider the sides 5, 7, and 8. We have:

tt tt52 72 25 49 74 tt82 64 tt tt

Since 74 is not equal to 64, the triangle does not satisfy the Pythagorean theorem, therefore, it is not a right triangle.

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Another approach is to consider the cosine rule, which is applicable to all triangles, not just right triangles. According to the law of cosines, for a triangle with sides (a), (b), and (c) opposite angles (A), (B), and (C) respectively, the following holds:

tt ttc2 a2 b2 - 2ab cosC tt tt

Here, let's identify the longest side as 8 (thus, (c 8)) and calculate the cosine of the largest angle:

tt tt82 52 72 - 2 * 5 * 7 * cosC tt64 25 49 - 70 cosC tt64 74 - 70 cosC tt70 cosC 74 - 64 tt70 cosC 10 ttcosC 10 / 70 ttcosC 1 / 7 tt tt

Since the cosine of angle (C) is not 0, the angle is not right. Instead, it can be determined that:

tt ttC ≈ arccos(1/7) ≈ 81.79° tt tt

This angle is not 90 degrees, confirming that the triangle is not a right triangle.

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For a triangle to be a right triangle, it must satisfy the Pythagorean theorem. The sides 5, 7, and 9 were also considered:

tt tt92 52 72 tt81 ≠ 25 49 tt81 ≠ 74 tt tt

Once again, the sum of the squares of the two shorter sides (5 and 7) does not equal the square of the longest side (9). Therefore, a triangle with sides 5, 7, and 9 is also not a right triangle.

Identifying Other Triangle Types

Since the given triangle is not a right triangle, let's use the cosine rule to determine the type of triangle:

t

Using the longest side for angle calculation:

tt ttcosC (52 72 - 82) / (2 * 5 * 7) ttcosC (25 49 - 64) / 70 ttcosC 10 / 70 ttcosC 1 / 7 ttC ≈ arccos(1/7) ≈ 81.79° tt tt

Since the angle C is greater than 90 degrees, the triangle is an obtuse triangle (one angle greater than 90 degrees).

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The advantages of finding the largest angle first:

tt ttt

It helps in identifying the type of triangle quickly—whether it is acute (all angles less than 90 degrees), right, or obtuse (one angle greater than 90 degrees).

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Using the largest angle, the Law of Sines can be applied to find the other angles without unnecessary complexity.

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Related Triangle Types and Formulas

There are other interesting types of triangles, such as Eisensteinian triangles, which are triangles with integer side lengths and one angle of 60 or 120 degrees. Here’s an example:

For the sides 3, 5, and 7, we can apply the Pythagorean approach:

t32 52 9 25 34 t72 49 t34 49 t

Welcome to a fascinating world of triangle types and theorems! Understanding the properties of triangles can help us solve various geometric problems gracefully.

For more information, you can refer to the Wikipedia articles on the Pythagorean theorem and triangle types.