Understanding Acute Triangles: Definition, Calculation, and Example

acute triangle

An acute triangle is a triangle in which all three angles are acute angles, meaning each angle is less than 90 degrees

An acute triangle is a triangle in which all three angles are acute angles, meaning each angle is less than 90 degrees. In other words, all angles in an acute triangle are strictly less than 90 degrees.

To better understand an acute triangle, let’s consider a specific example.

Suppose we have a triangle with sides of lengths 5 cm, 7 cm, and 9 cm.

To determine if this triangle is acute, we need to examine its angles.

Using the Law of Cosines, we can find the angles of the triangle.

Let’s denote the sides of the triangle as a, b, and c. The corresponding angles will be denoted as A, B, and C.

In our example, we have a = 5 cm, b = 7 cm, and c = 9 cm.

Using the Law of Cosines, we can calculate the cosines of the angles as follows:

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)

Plugging in the values, we get:

cos(A) = (7^2 + 9^2 – 5^2) / (2 * 7 * 9) = 0.877
cos(B) = (5^2 + 9^2 – 7^2) / (2 * 5 * 9) = 0.572
cos(C) = (5^2 + 7^2 – 9^2) / (2 * 5 * 7) = 0.813

To find the angles A, B, and C, we can use the inverse cosine function (cos^-1) on the calculated values:

A = cos^-1(0.877) ≈ 28.07 degrees
B = cos^-1(0.572) ≈ 56.63 degrees
C = cos^-1(0.813) ≈ 95.30 degrees

Since all three angles A, B, and C are less than 90 degrees, this triangle is an acute triangle.

In summary, an acute triangle is a triangle in which all three angles are less than 90 degrees. By using the Law of Cosines or other trigonometric functions, we can determine the angles of a given triangle and classify it as acute if all angles satisfy the condition.

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