Question

A triangle has leg lengths of 8, 12 and 14. Classify the triangle
Acute
Obtuse
Right

Answer 1

To classify the triangle, we can use the Pythagorean Theorem. If the sum of the squares of the two shorter sides is equal to the square of the longest side, then the triangle is classified as a right triangle.

In this case, the triangle has leg lengths of 8 and 12. We can check if it is a right triangle by adding their squares and comparing it to the square of the longest side:

8^2 + 12^2 = 64 + 144 = 208
14^2 = 196

Since 208 is greater than 196, we can conclude that the triangle is not a right triangle.

To further classify the triangle, we can use the Law of cosines. For any triangle with sides a, b, and c and opposite angles A, B, and C respectively, the Law of cosines states:

c^2 = a^2 + b^2 - 2ab * cos(C)

Let's calculate it:

14^2 = 8^2 + 12^2 - 2(8)(12) * cos(C)
196 = 64 + 144 - 192 * cos(C)
196 = 208 - 192 * cos(C)
192 * cos(C) = 12
cos(C) = 12/192
cos(C) = 1/16

Now, we can compare the value of cos(C) to classify the triangle:

If cos(C) is greater than 0, then the triangle is acute.
If cos(C) is less than 0, then the triangle is obtuse.

In this case, cos(C) is positive (1/16 > 0), which means that the triangle is acute.

Therefore, the triangle with leg lengths of 8, 12, and 14 is classified as an acute triangle.