Suppose a triangle has side lengths of 6, 7, and 10. What best describes it? Acute, right, obtuse, or not enough information?
To determine the type of triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's check if the given triangle with side lengths of 6, 7, and 10 satisfies the Pythagorean theorem:
The longest side (10) squared is equal to the sum of the squares of the other two sides (6 squared + 7 squared):
10^2 = 6^2 + 7^2
100 = 36 + 49
100 = 85
Since 100 is not equal to 85, the given triangle does not satisfy the Pythagorean theorem. Therefore, the triangle is not a right triangle.
Now, if a triangle is not a right triangle, it can either be acute or obtuse. To determine this, we can compare the squares of the two shorter sides.
Let's calculate the squares of the two shorter sides:
6^2 = 36
7^2 = 49
Since 36 + 49 = 85, which is less than the square of the longest side (10^2 = 100), we can conclude that the triangle with side lengths 6, 7, and 10 is an obtuse triangle.
Therefore, the best description for the given triangle is obtuse.
To determine the type of the triangle, we need to examine the angles of the triangle. In this case, we can use the Pythagorean theorem to determine whether it is a right triangle or not.
The Pythagorean theorem states that in a right triangle, the square of the length of the longest side, also known as the hypotenuse, is equal to the sum of the squares of the other two sides.
In this triangle, with side lengths of 6, 7, and 10, we can see that 10 is the longest side. We can test whether this triangle is right or not by checking if 10² is equal to 6² + 7²:
10² = 100
6² + 7² = 36 + 49 = 85
Since 100 is equal to 85, the triangle satisfies the Pythagorean theorem, meaning it is a right triangle.
Therefore, the best description for the given triangle with side lengths 6, 7, and 10 is "right triangle."
It is a scalene triangle. One angle is obtuse, and no two are equal.
The angle opposite the longest side is C. According to the Law of Cosines,
c^2 = a^2 + b^2 -2ab cos C.
Since c^2 is greater than a^2 + b^2, cos C is negative, so C is obtuse ( >90 degrees)