Problem 61 - Precalculus : Building Concepts and Connections

Question

If you are given two sides of a triangle and the included angle (SAS), how can you tell whether the triangle has a right angle if the included angle is acute?

Step-by-Step Solution

Answer

To determine whether the triangle is a right triangle, apply the Law of Cosines and see if \(c^2 = a^2 + b^2\) holds true. If it does, the triangle is a right triangle because the square of one side equals the sum of the squares of the other two sides.

Step 1: Understand the Law of Cosines

The Law of Cosines states that in any given triangle with sides of length a, b and c and the angle \(\theta\) included between sides a and b, the following holds: \(c^2 = a^2 + b^2 - 2ab \cos(\theta)\). This formula gives the square of the side across from the given angle (\(\theta\)) in terms of the squares of the other sides and the cosine of the given angle. Here, 'c' is the side which needs to be determined whether it is a hypotenuse or not.

Step 2: Apply the Law of Cosines Considering the Given Angle is Acute

In an acute angle, the cosine function is positive. Hence if we apply the Law of Cosines for this scenario, we will get \(c^2 = a^2 + b^2 - 2ab \cos(\theta)\), where \(\theta\) is the acute angle.

Step 3: Comparing with the Property of Right Triangle

After getting the value of \(c^2\) using the Law of Cosines, we need to recall the main property of right triangles: in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides i.e., \(c^2 = a^2 + b^2\). In this equation, if 'c' is the longest side then the triangle is a right triangle.

Step 4: Final judgement

Compare the result obtained in step 2 with the right triangle condition i.e., \(c^2 = a^2 + b^2\). If the results match, then it indicates that the triangle is a right triangle. However, if the results do not match, it means that the triangle is not a right one as 'c' is not acting like a hypotenuse in this scenario. Hence, determining whether the triangle is a right triangle depends on the results obtained from the Law of Cosines, compared with the right triangle condition.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle exactly equal to 90 degrees. This 90-degree angle is known as the right angle. The side opposite this right angle is termed the hypotenuse, and it is always the longest side. The other two sides are referred to as the legs.

When you know a triangle is right, you can make use of properties specific to right triangles, like the Pythagorean theorem. This theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse:

• If the legs of the triangle are denoted as 'a' and 'b', and the hypotenuse as 'c', then the relationship is: \[c^2 = a^2 + b^2 \]

To identify if a triangle with given sides forms a right triangle, you can compare this relationship to the values you have. If the largest side satisfies this equation, then the triangle is a right triangle.

Acute Angle

An acute angle is an angle that is less than 90 degrees but more than 0 degrees. In a triangle, an acute angle means that each of the interior angles is less than 90 degrees.

When dealing with acute angles in the context of triangles, particularly when using the Law of Cosines, you should note that the cosine of an acute angle is positive. This characteristic plays a significant role in various triangle computations, especially when determining side lengths.

• The Law of Cosines, when applied with an acute angle, helps you calculate the length of a side if the other two sides and the included angle are known. For triangle sides 'a' and 'b' with an acute included angle \(\theta\), the length of the third side 'c' can be found as: \[c^2 = a^2 + b^2 - 2ab \, \cos(\theta)\]

The positivity of cosine in an acute angle ensures that the term \(-2ab \cos(\theta)\) subtracts a small positive value, slightly reducing the calculated length of 'c.' This property is key when comparing if the resulting side 'c' fits the requirement of being the hypotenuse of a right triangle.

SAS (Side-Angle-Side)

SAS refers to a scenario in which two sides of a triangle and the included angle between them are known. This specific arrangement allows you to use the Law of Cosines effectively to determine the missing side of the triangle.

By using SAS, you are equipped to solve for one of the missing sides of a triangle when:

• The lengths of two sides (say 'a' and 'b') are provided.
• The measure of the angle \(\theta\) between these sides is provided.

The Law of Cosines formula comes into play here:

• \[c^2 = a^2 + b^2 - 2ab \cos(\theta)\]

This equation allows calculation of 'c', the side opposite the given angle \(\theta\). It is crucial in determining whether a triangle is right, especially when the included angle \(\theta\) is acute, by comparing it to the classic right triangle relationship.

In essence, SAS isn't just a set of given values; it's a powerful approach that opens doors to calculating unknown dimensions of a triangle, helping you assess its nature effectively.