Video: Identifying the Type of an Angle in a Triangle Using the Triangle Inequality Theorem

In triangle 𝐴𝐡𝐢, (𝐴𝐡)² < (𝐡𝐢)² + (𝐴𝐢)². What type of angle is 𝐢?

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Video Transcript

In triangle 𝐴𝐡𝐢, 𝐴𝐡 squared is less than 𝐡𝐢 squared plus 𝐴𝐢 squared. What type of angle is 𝐢?

If we consider any triangle 𝐴𝐡𝐢, there are three possibilities for angle 𝐢. The first diagram shows that angle 𝐢 is a right angle. The second diagram shows that angle 𝐢 is an acute angle, as it is less than 90 degrees. The third diagram shows that angle 𝐢 is an obtuse angle, as it is greater than 90 degrees but less than 180 degrees.

We need to consider the relationship between the lengths of sides in triangles and decide which one corresponds to 𝐴𝐡 squared is less than 𝐡𝐢 squared plus 𝐴𝐢 squared. Pythagoras’s theorem states that, in any right-angled triangle, 𝐴 squared plus 𝐡 squared is equal to 𝐢 squared, where 𝐢 is the longest side of the triangle, known as the hypotenuse. This means that, in our first diagram, 𝐴𝐡 squared is equal to 𝐡𝐢 squared plus 𝐴𝐢 squared.

We can therefore rule out right angle for angle 𝐢. When angle 𝐢 gets larger, as in the third diagram, the length 𝐴𝐡 also becomes larger. We can therefore say that when angle 𝐢 is obtuse, 𝐴𝐡 squared is greater than 𝐡𝐢 squared plus 𝐴𝐢 squared. We can therefore also rule out an obtuse angle.

In the second diagram, when angle 𝐢 becomes smaller, the length 𝐴𝐡 also becomes smaller. This means that when angle 𝐢 is acute, 𝐴𝐡 squared is less than 𝐡𝐢 squared plus 𝐴𝐢 squared. This inequality corresponds to the one in the question. We can therefore conclude that angle 𝐢 is acute.

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