WEBVTT
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Find 𝑥 in the right triangle shown.
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Looking at the information we’ve been given, we note, first of all, that this triangle is a right triangle.
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It includes a right angle.
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And we’ve been given the lengths of two of its sides.
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They are eight units and 15 units.
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𝑥 represents the length of the third side of this right triangle.
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And from its position, directly opposite the right angle, we note that 𝑥 is the hypotenuse of this triangle.
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As we’ve been given the lengths of two sides in a right triangle and we wish to calculate the third, this is exactly the setup we need in order to apply the Pythagorean theorem.
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This tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
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So we’ll begin by writing down what the Pythagorean theorem tells us about this triangle in particular.
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The two shorter sides are eight units and 15 units.
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So the sum of the squares of the two shorter sides is eight squared plus 15 squared.
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This is then equal to the square of the hypotenuse.
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And as the hypotenuse of our triangle is 𝑥, we now have the equation eight squared plus 15 squared is equal to 𝑥 squared.
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So by considering what the Pythagorean theorem tells us about this triangle in particular, we have an equation we can solve in order to determine the value of 𝑥.
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You may prefer to swap the two sides of the equation around so that 𝑥 is on the left-hand side, although this isn’t entirely necessary.
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Now that we formed our equation, we’re going to solve it by first evaluating eight squared and 15 squared.
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This gives 𝑥 squared equals 64 plus 225, which simplifies to 𝑥 squared equals 289.
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The next step in solving this equation is to take the square root of each side because the square root of 𝑥 squared will give 𝑥.
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Now usually, when we solve an equation by square rooting, we must remember to take plus or minus the square root.
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But here 𝑥 has a physical meaning; it’s the length of a side in a triangle.
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So it must take a positive value.
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We therefore write 𝑥 equals just the positive square root of 289.
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289 is in fact a square number, and its square root is 17.
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So we found the value of 𝑥.
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𝑥 is equal to 17.
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Now, we should always perform a quick sense check of our answer by comparing the value we found with the other two sides in the triangle.
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Remember, 𝑥 represents the hypotenuse, which is the longest side in this right triangle.
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So our value for 𝑥 needs to be bigger than the lengths of the two other sides.
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Our value is 17 and the two other sides are 15 and eight.
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So our answer does make sense.
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Now, in fact, this triangle is an example of a special type of right triangle, called a Pythagorean triple.
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This is a right triangle in which all three of the side lengths are integers.
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The most well-known Pythagorean triple is the three-four-five triangle as three squared plus four squared is equal to five squared.
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You may well encounter Pythagorean triples when working without a calculator.
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So it’s a good idea to be familiar with some of the most common ones.
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By applying the Pythagorean theorem then, we’ve found the value of 𝑥 in the right triangle shown is 17.