Video: Identifying the Type of a Triangle by Applying Triangular Inequalities

Suppose that, in โ–ณ๐ด๐ต๐ถ, ๐ด๐ต = 13 cm, ๐ต๐ถ = 36 cm and ๐ด๐ถ = 35 cm. What kind of triangle is this, in terms of its angles? [A] It is a right-angled triangle. [B] It is an acute-angled triangle. [C] It is an obtuse-angled triangle.

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Video Transcript

Suppose that, in triangle ๐ด๐ต๐ถ, ๐ด๐ต equals 13 centimeters, ๐ต๐ถ equals 36 centimeters, and ๐ด๐ถ equals 35 centimeters. What kind of triangle is this in terms of its angles? It is a right-angled triangle, it is an acute-angled triangle, or it is an obtuse-angled triangle.

So weโ€™ve been given the three side lengths of a triangle and asked to determine what type of angles it has. In order to do this, we need to look at the relationship between the squares of these three sides.

From the information in the question, we can see that ๐ต๐ถ is the longest side. If the triangle is a right-angled triangle, then the Pythagorean theorem will hold true for its lengths, which means that the square of the longest side, in this case ๐ต๐ถ, will be equal to the sum of the squares of the two shorter sides, ๐ด๐ต and ๐ด๐ถ.

So the triangle will be right-angled if ๐ต๐ถ squared is equal to ๐ด๐ต squared plus ๐ด๐ถ squared. If instead itโ€™s true that ๐ต๐ถ squared is less than the sum of the squares of the other two sides, then the triangle will be acute-angled.

The final possibility is that ๐ต๐ถ squared will be greater than ๐ด๐ต squared plus ๐ด๐ถ squared. And in this case, this will mean that the triangle is obtuse-angled. So letโ€™s evaluate the squares of the three sides in order to determine what type of triangle we have.

๐ต๐ถ squared is equal to 36 squared, which is 1296. ๐ด๐ต squared plus ๐ด๐ถ squared is 13 squared plus 35 squared, which evaluated is 1394. Now letโ€™s compare these values. 1296 is less than 1394. So ๐ต๐ถ squared is less than ๐ด๐ต squared plus ๐ด๐ถ squared. Therefore, we can conclude that this triangle is an acute-angled triangle.

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