Understanding Acute Triangles: Definition, Characteristics, and Determination

acute triangle

An acute triangle is a type of triangle in which all three angles are less than 90 degrees

An acute triangle is a type of triangle in which all three angles are less than 90 degrees. In other words, the angles of an acute triangle are all “acute” or smaller than a right angle.

Key characteristics of an acute triangle:

1. Angle Measurements: All three angles of an acute triangle are less than 90 degrees.
2. Side Lengths: The lengths of the sides can vary. There is no specific requirement for side lengths in an acute triangle.
3. Perimeter: The perimeter of an acute triangle is the sum of the lengths of all three sides.

To determine if a triangle is acute, given its side lengths, you need to check if all three angles are less than 90 degrees. You can do this by using the Law of Cosines or the Law of Sines.

Example:

Let’s consider a triangle with side lengths of 5 cm, 8 cm, and 10 cm. We want to determine if this triangle is acute.

Step 1: Use the Law of Cosines to find the largest angle in the triangle.

Using the formula: c^2 = a^2 + b^2 – 2ab * cos(C), where a, b, and c are the side lengths and C is the angle opposite side c.

Substituting the values, we have:
10^2 = 5^2 + 8^2 – 2(5)(8) * cos(C)
100 = 25 + 64 – 80cos(C)
100 = 89 – 80cos(C)
80cos(C) = -11
cos(C) = -11/80

Step 2: Find the largest angle in the triangle.

Since cos(C) = -11/80, we need to find the inverse cosine of -11/80. Using a calculator, we find that cos^(-1)(-11/80) ≈ 94.9 degrees.

Therefore, the largest angle in this triangle is approximately 94.9 degrees.

Step 3: Check if all three angles are less than 90 degrees.

Since the largest angle in the triangle is approximately 94.9 degrees, we can conclude that not all angles in this triangle are less than 90 degrees. Hence, the triangle is not an acute triangle.

Remember, an acute triangle is defined by the measure of its angles, with all three angles being less than 90 degrees.

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