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In this video, we will learn how to use the Pythagorean theorem, or Pythagoras’ theorem as you may know it, to find the lengths of sides in right triangles.
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This is hugely useful as right triangles can be used to model a lot of different physical scenarios.
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And they’re often involved in problems involving area.
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The Pythagorean theorem is one of those really well-known bits of maths.
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And many people will still remember its name from their school days even if they can no longer remember what the theorem itself actually says.
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So, what does the Pythagorean theorem say?
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Well, the Pythagorean theorem is all about the special relationship that exists between the lengths of the three sides in a right triangle, that is, a triangle that includes a right angle.
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Remember, we call the longest side of a right triangle, which is always the side directly opposite the right angle, the hypotenuse.
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The Pythagorean theorem then says this.
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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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Often, we use the letters 𝑎 and 𝑏 to represent the two shorter sides or legs of the right triangle.
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And we use the letter 𝑐 to represent the hypotenuse, in which case the Pythagorean theorem can be expressed as 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared.
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But it’s important to remember what the theorem itself is saying, not just to learn this equation.
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In pictorial terms, what the Pythagorean theorem is telling us is that if were to draw a square on each side of a right triangle, then the sum of the areas of two smaller squares would equal the area of the largest square.
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That’s the square on the hypotenuse.
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There are lots of different ways to prove the Pythagorean theorem, but one of the nicest, in my opinion, is a method called Perigal’s dissection.
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We won’t go into detail here.
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But it involves chopping the two smaller squares up and rearranging the pieces to fit exactly inside the larger square, as you can see in the diagram here.
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If you like, you could try this yourself by reproducing this diagram on a piece of paper.
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Let’s now have a look at some examples of how we can apply the Pythagorean theorem.
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We’ll begin by considering example of how to use the theorem to find the hypotenuse of a right triangle.
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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.
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In our next example, we’ll see how to apply the Pythagorean theorem to find the length of one of the two shorter sides in a right triangle.
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Find 𝑥 in the right triangle shown.
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So we have a right triangle.
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And we’re asked to find the value of 𝑥, which represents the length of one of the triangle sides.
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We’ve been given the lengths of the other two sides.
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So we have exactly the right set of information 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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Now, before applying the Pythagorean theorem, we must be very careful to make sure we correctly identify which side of the triangle is the hypotenuse.
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And remember, it’s always the side directly opposite the right angle.
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So in this case, the hypotenuse of the triangle is 13 units.
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The side we’ve been asked to find, length 𝑥, is one of the two shorter sides or legs of this right triangle.
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So the first thing we do then is to write down what the Pythagorean theorem tells us about this particular triangle.
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The two shorter sides are 𝑥 and 12.
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So the sum of their squares will be 𝑥 squared plus 12 squared.
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The hypotenuse of the triangle is 13 units.
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So if the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, we have the equation 𝑥 squared plus 12 squared equals 13 squared.
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Now that we’ve formed our equation, we’re going to solve it to determine the value of 𝑥.
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First, we evaluate 12 squared and 13 squared, giving 𝑥 squared plus 144 equals 169.
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We want to leave 𝑥 or 𝑥 squared initially on its own on the left-hand side of the equation.
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So the next step is to subtract 144 from each side.
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On the left-hand side, 𝑥 squared plus 144 minus 144 just leaves 𝑥 squared.
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And on the right-hand side, 169 minus 144 is 25.
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The final step is to take the square root of each side of the equation, remembering we only need to take the positive square root as 𝑥 represents a length.
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So it must have a positive value.
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𝑥 is therefore equal to the square root of 25.
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And as 25 is a square number, its square root is an integer; it’s simply five.
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So we found the value of 𝑥.
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𝑥 is equal to five.
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Now, in fact, this triangle is an example of a Pythagorean triple.
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That’s a right triangle in which all three side lengths are integers.
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We should also perform a quick check of our answer.
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Remember, we were looking to calculate one of the shorter sides of this triangle.
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So our value for 𝑥 must be less than the length we were given for the hypotenuse.
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Five is certainly less than 13.
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So our answer makes sense.
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So by applying the Pythagorean theorem, we’ve solved this problem.
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The value of 𝑥 is five.
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We must make sure we’re really careful when setting up our equation.
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And we need to be sure before we begin whether we’ve been asked to find the length of one of the shorter sides or the length of the hypotenuse.
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So we’ve now seen one example of calculating the length of the hypotenuse and one example of calculating the length of one of the shorter sides.
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The Pythagorean theorem is really useful because it helps us answer lots of different types of practical problems.
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So we’ll now consider a couple of examples with a greater focus on problem solving.
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Determine the diagonal length of the rectangle whose length is 48 centimetres and width is 20 centimetres.
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Now, we haven’t been given a diagram for this question.
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So it’s always a good idea to begin by drawing our own.
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We have a rectangle with a length of 48 centimetres and a width of 20 centimetres.
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The length we’ve been asked to calculate is the diagonal of this rectangle.
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That’s the line that joins opposite corners together.
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We can use the letter 𝑑 to represent this unknown length.
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Now we know that all the interior angles in a rectangle are 90 degrees.
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So, in fact, this problem isn’t just about rectangles.
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It’s also about right triangles, that is, the triangle formed by the rectangle’s length, its width, and this diagonal.
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Looking at the lower triangle in our diagram, we can see that we’ve been given the lengths of two of its sides — they’re 20 centimetres and 48 centimetres — and asked to calculate the length of its third side.
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And as this is a right triangle, we’re going to be able to do this by applying 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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Now, before we try to apply the Pythagorean theorem, we must identify which of the three sides we’ve been asked to calculate.
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Remember, the hypotenuse is always the side directly opposite the right angle.
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So the side we’re looking to find is the hypotenuse of a triangle.
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We then ask ourselves, “what does the Pythagorean theorem tell us, not just in general, but about this triangle specifically?”
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Well, as the two shorter sides are 48 and 20 centimetres, the sum of their squares is 48 squared plus 20 squared.
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And the square of the hypotenuse is 𝑑 squared.
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So we have the equation 48 squared plus 20 squared equals 𝑑 squared.
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We can of course swap the two sides of the equation round if we prefer to have 𝑑 squared on the left-hand side.
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So by applying the Pythagorean theorem, we’ve formed an equation, which we can now solve in order to determine the value of 𝑑.
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First, we evaluate 48 squared and 20 squared and then add these values together to give 𝑑 squared is equal to 2704.
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The final step in solving this equation is to take the square root of each side, giving 𝑑 equals the square root of 2704.
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Now, 2704 is in fact a square number although probably not one that you are overly familiar with.
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Its square root is simply 52.
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So we have that 𝑑 is equal to 52.
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The diagonal length of this rectangle then is 52 centimetres.
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Now, we should perform a quick sense check of our answer.
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Remember, 𝑑 was the hypotenuse of this triangle.
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It’s supposed to be the longest side.
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So we need to check that our value does make sense.
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Well, 52 is indeed greater than each of the other side lengths.
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So it’s a sensible value for the hypotenuse of this triangle.
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So we’ve completed the problem.
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The key stage in this question was to first draw our own diagram.
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And once we did, we saw that this problem wasn’t just about rectangles.
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It was in fact about right triangles.
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And hence, we could solve it by applying the Pythagorean theorem.
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Let’s now consider one final example involving points plotted on a coordinate grid.
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A triangle has vertices of the points 𝐴 four, one; 𝐵 six, two; and 𝐶 two, five.
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Work out the lengths of the sides of the triangle.
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Give your answers as surds in their simplest form.
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And secondly, is this triangle a right triangle?
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Let’s begin by sketching this triangle on a coordinate grid.
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We absolutely don’t need to plot this triangle accurately.
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We aren’t going to be measuring the lengths of any of the lines.
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We just want to sketch it using the approximate position of these three points relative to one another.
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So the triangle looks a little something like this.
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Now, from our sketch, it looks possible that this could be a right triangle with the right angle at 𝐴.
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But we can’t confirm this from our sketch.
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Let’s consider the first part of the question.
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We need to find the lengths of the three sides of the triangle.
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And we’ll begin by finding the length of the side 𝐴𝐵.
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We can sketch in a right triangle below this line using 𝐴𝐵 as its hypotenuse.
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We can also work out the lengths of the other two sides in this triangle.
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The horizontal side will be the difference between the 𝑥-values at its endpoints.
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That’s the difference between six and four, which is two.
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And the vertical side will be the difference between the 𝑦-values at its endpoints.
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That’s the difference between two and one, which is one.
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As we now have the lengths of two sides in a right triangle and we wish to calculate the length of the third side, we can apply the Pythagorean theorem, which 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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Remember, 𝐴𝐵 is the hypotenuse.
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So we have that 𝐴𝐵 squared is equal to one squared plus two squared.
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One squared is one and two squared is four.
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So adding these values together, we have that 𝐴𝐵 squared is equal to five.
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To find the length of 𝐴𝐵, we need to square root each side of this equation.
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And remember at this point, we’ve been told to give our answer as a surd.
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So we have that 𝐴𝐵 is equal to root five.
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We can find the lengths of the other two sides of the triangle in the same way.
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We sketch in a right triangle below the line 𝐵𝐶.
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And we see that it has a horizontal side of four units and a vertical side of three units.
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𝐵𝐶 is the hypotenuse of this triangle.
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So applying the Pythagorean theorem, we have that 𝐵𝐶 squared is equal to three squared plus four squared.
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That’s nine plus 16, which is equal to 25.
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𝐵𝐶 is therefore equal to the square root of 25, which is simply the integer five.
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In the same way, 𝐴𝐶 is the hypotenuse of a right triangle with shorter sides of two and four units.
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So 𝐴𝐶 is equal to the square root of 20, which simplifies to two root five.
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So we’ve answered the first part of the question.
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And now we need to determine whether this triangle is a right triangle.
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Well, if it is, then the Pythagorean theorem will hold for its three side lengths.
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Now we suspect it that the right angle was at 𝐴, which would make 𝐵𝐶 the hypotenuse of the triangle if it is indeed a right triangle.
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We therefore want to know whether 𝐵𝐶 squared is equal to 𝐴𝐵 squared plus 𝐴𝐶 squared.
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Well, we can in fact use the squared side lengths.
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We know that 𝐵𝐶 squared is 25.
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We know that 𝐴𝐵 squared is five.
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And we know that 𝐴𝐶 squared is 20.
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So is it true that 25 is equal to five plus 20?
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Yes, of course, it’s true, which means that the Pythagorean theorem holds for this triangle.
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And therefore, it is indeed a right triangle.
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So we’ve completed the problem.
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We have the three side lengths.
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𝐴𝐵 equals root five, 𝐵𝐶 equals five, and 𝐴𝐶 equals two root five.
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And we’ve determined that the triangle is a right triangle.
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Let’s now summarise what we’ve seen in this video.
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The Pythagorean theorem 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, which we may often see written as 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared.
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The first step in any problem should be to write down what the Pythagorean theorem tells us about the particular triangle in this problem.
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That is, we form an equation.
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We then solve our equation, which will involve square rooting.
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Finally, we should always check our answer by making sure that the value we’ve calculated makes sense in relation to the lengths of the other two sides.