A triangle has side lengths 10,15,and 7.Is the triangle acute,obtuse,or right?Explain.
if ABC is a right triangle (B = 90°),
7^2 + 10^2 = 15^2
but 149 < 225, so B < 90° and ABC is acute
To determine whether a triangle is acute, obtuse, or right, we need to examine the relationship between the lengths of its sides.
In this case, we have a triangle with side lengths 10, 15, and 7.
To determine if it is acute, obtuse, or right, we need to apply the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
The square of the longest side is called the hypotenuse. So, let's find the longest side first. Comparing the side lengths, we see that 15 is the largest side.
Now, let's check if this triangle is a right triangle by using the Pythagorean theorem:
If 15^2 = 10^2 + 7^2, then it is a right triangle.
Calculating these values, we find:
15^2 = 225
10^2 + 7^2 = 100 + 49 = 149
Since 225 is not equal to 149, the triangle does not satisfy the Pythagorean theorem and is not a right triangle.
Therefore, the triangle with side lengths 10, 15, and 7 is either acute or obtuse.
To determine whether it is acute or obtuse, we need to analyze the angles of the triangle.
Using the Law of Cosines, we can determine the angles of the triangle by using the formula:
cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)
cos(B) = (a^2 + c^2 - b^2) / (2 * a * c)
cos(C) = (a^2 + b^2 - c^2) / (2 * a * b)
Substituting the values a = 10, b = 15, and c = 7 into the formulas, we can calculate the cosines of each angle.
cos(A) = (15^2 + 7^2 - 10^2) / (2 * 15 * 7) = 0.57
cos(B) = (10^2 + 7^2 - 15^2) / (2 * 10 * 7) = -0.35
cos(C) = (10^2 + 15^2 - 7^2) / (2 * 10 * 15) = 0.9
To determine the type of the triangle, we need to examine the angle measures corresponding to these cosine values.
Since all of the cosine values are positive, we can conclude that all angles of the triangle are acute angles.
Therefore, the triangle with side lengths 10, 15, and 7 is an acute triangle.