Question Video: Solving for the Hypotenuse of a Right-Angled Triangle with Non-integer Solutions | Nagwa Question Video: Solving for the Hypotenuse of a Right-Angled Triangle with Non-integer Solutions | Nagwa

Question Video: Solving for the Hypotenuse of a Right-Angled Triangle with Non-integer Solutions Mathematics • First Year of Preparatory School

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In the following figure, find the length of line segment 𝐴𝐢.

02:12

Video Transcript

In the following figure, find the length of line segment 𝐴𝐢.

Looking at the figure, we see that we have been given a right triangle in which the lengths of two of the sides are known. 𝐴𝐡 is 116 centimeters, and 𝐡𝐢 is 121.8 centimeters. We’re asked to find the length of the line segment 𝐴𝐢, which is the third side of this triangle.

Now, we should recall that whenever we know the lengths of two sides in a right triangle and want to calculate the length of the third side, we can do this by applying the Pythagorean theorem. This states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. If we label the two shorter sides as having lengths π‘Ž and 𝑏 and the hypotenuse as having length 𝑐, then this can be expressed as π‘Ž squared plus 𝑏 squared equals 𝑐 squared.

In our triangle, the two shorter sides are the sides whose lengths we’ve been given and the hypotenuse is side 𝐴𝐢, because it’s directly opposite the right angle. So, by the Pythagorean theorem, we have that 𝐴𝐢 squared is equal to 121.8 squared plus 116 squared. To solve this equation for 𝐴𝐢, we first evaluate the squares and then find their sum, giving 𝐴𝐢 squared equals 28291.24. 𝐴𝐢 is then equal to the square root of this value. We take only the positive value here as 𝐴𝐢 is a length and so must be positive. Evaluating this on a calculator gives 168.2.

Hence, by applying the Pythagorean theorem to calculate the length of the hypotenuse of this right triangle, we’ve found that the length of the line segment 𝐴𝐢 is 168.2 centimeters.

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