Question

Which classification best represents a triangle with side lengths 6 cm, 10 cm, and 12 cm? acute, because 62 + 102 < 122 acute, because 6 + 10 > 12 obtuse, because 62 + 102 < 122 obtuse, because 6 + 10 > 12

Answer 1

Using the Pythagorean theorem, we find that the sum of the squares of the two shorter sides of the triangle (6 cm and 10 cm) is less than the square of the longest side (12 cm). Hence, the triangle is obtuse because 6² + 10² < 12².

Step-by-step explanation:

To classify a triangle whose sides measure 6 cm, 10 cm, and 12 cm, we can use the Pythagorean theorem which relates the lengths of a right triangle's legs (a and b) to its hypotenuse (c), by the formula a² + b² = c². In our case, we need to check if the squares of the two shorter sides add up to the square of the longest side. We calculate:

• 6² + 10² = 36 + 100 = 136
• 12² = 144

Since 136 is less than 144, 62 + 102 < 122 is true, and the triangle is obtuse because the sum of the squares of the lengths of the two shorter sides is less than the square of the length of the longest side.

Answer 2

The Pythagoras theorem states that
the sum of squares of the shorter sides (legs) of a right triangle equals the square of the third side.

A corollary from the same theorem helps us solve this problem:
If the sum of the squares of the shorter sides of a triangle is greater than the square of the third side, the included angle is acute. ..... (case 1)
Conversely, if the sum of the squares of the shorter sides of a triangle is less than the square of the third side, the triangle is obtuse. .....(case 2)

Here we have
6^2+10^2 = 36+100=136 <12^2=144
Therefore case 2 applies, and the triangle is obtuse.