A Proof of the Pythagorean Theorem
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Basics on the topic A Proof of the Pythagorean Theorem
After this lesson you will be able to prove the Pythagorean Theorem using a “square within a square”.
The lesson begins by recalling the definition of the Pythagorean Theorem. It leads to calculating the area of the large square and setting it equal to the area of the shapes it is composed of. It concludes with the equation for the Pythagorean theorem.
Learn how to prove the Pythagorean Theorem by helping Maurice complete his homework!
This video includes key concepts, notation, and vocabulary such as: Pythagorean Theorem (given a right triangle, the sum of the squares of legs equals the square of the hypotenuse); Angle Sum Theorem (the sum of the interior angles of a triangle is 180 degrees); straight angle (an angle whose measure is 180 degrees); area of a triangle (one half base times height); area of a square (the square of the base); binomial (two terms); squaring a binomial (multiplying the binomial by itself); and expanding a binomial (multiplying two binomials).
Before watching this video, you should already be familiar with rigid motion transformations, Angle Sum Theorem, expanding a binomial, and the area of a triangle and a square.
After watching this video, you will be prepared to problem solve using the Pythagorean Theorem.
Common Core Standard(s) in focus: 8.G.B.6 A video intended for math students in the 8th grade Recommended for students who are 13-14 years old
Transcript A Proof of the Pythagorean Theorem
Maurice is trying to do his homework on applying the Pythagorean Theorem while watching his little sister, Sina when suddenly Eureka! The arrangement of Sina's blocks gives Maurice an idea for a proof of the Pythagorean Theorem! If we prove that the Pythagorean Theorem is true for a right triangle with variable side lengths, that will mean it will be true for right triangles of all side lengths. Remember, the Pythagorean Theorem tells us that, given a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. We write this algebraically as 'a' squared plus 'b' squared equals 'c' squared. This is the theorem that we want to prove using logical statements and properties we already know. Once we prove the Pythagorean Theorem we can use it to find a missing side length. Notice that the blocks, Sina is playing with, show a large square that is actually made up of 4 right triangles and a smaller square. Each side of the large square is divided into lengths of 'a' and 'b'. To prove that the Pythagorean Theorem is true, we will calculate the area of the large square using its side lengths and set it equal to the area of the shapes it is composed of. Recall, the area of a square is just its side length squared. What is the side length of the large square? It's 'a' plus 'b'. That means the area of the large square is the quantity 'a' plus 'b' squared. Remember, squaring means multiplying by itself, so that's just 'a' plus 'b' times 'a' plus 'b'. Notice that we are multiplying two binomials. Therefore we need to distribute, which means to multiply each term in the first expression by the terms in the second. That gives us 'a' squared plus 'ab' plus 'ba' plus 'b' squared. We can simplify by combining like terms. That is 'ab' plus 'ba' is '2ab'. So, the total area of the large square is 'a' squared plus '2ab' plus 'b' squared. We can find the area of the large square by also summing up the areas of the triangles and the small square. Now, let's find the area of the triangles. Are these triangles congruent? Placing them on top of each other shows us that the triangles are in fact the same. So, we can just start by finding the area of one triangle to find the area of them all. What is the formula for the area of a triangle, again? Recall, area of a triangle is one-half the base times the height. We can substitute 'a' for the base and 'b' for the height resulting in one-half 'ab.' This is just the area for one triangle so, the total area of all four triangles is 4 times one-half 'ab'. Simplifying shows us the area of all the triangles can be expressed as '2ab'. Now let's look at the smaller square. Actually, how do we know that this is truly a square? Remember to be a square the shape must have four equal sides and four right angles. Because the four triangles are congruent, these four segments are congruent. We'll call the length of each side, 'c'. So, it has four equal sides but what about the right angles? In order to determine the value of these angles let's recall the angle sum theorem that states, the sum of the measures of the interior angles of a triangle is 180 degrees. Since we already know the triangle has one angle measure of 90 degrees it means these two acute angles... must sum to 90 degrees. Because all the triangles are congruent it means their corresponding angles are also congruent. Therefore, these two angles also sum to 90 degrees. And since this is a straight angle, measuring 180 degrees that means this angle of the inside shape must be 90 degrees. This inside shape has 4 right angles, meaning it is in fact a square. Since it is a square, we can quickly calculate its area by finding the side length squared. It has a side length of 'c.' So, it's area is 'c' squared. Remember that the area of the large square is equal to the area of the four triangles plus the area of the small square. The area of the large square is 'a' squared plus '2ab' plus 'b' squared. The area of the four triangles is '2ab'... and the area of the small square is 'c' squared. Notice that '2ab' occurs on both sides of the equation. So, we can use the subtraction property of equality to subtract '2ab' from both sides. This leaves us with 'a' squared plus 'b' squared equals 'c' squared. A' and 'b' are the legs of a right triangle, and 'c' is the hypotenuse. This proves that the Pythaogrean Theorem is true for ALL right triangles. Let's review our proof of the Pythagorean Theorem. We started with a shape composed of a square within a square. We calculated the area of the large square and set it equal to the area of the shapes it is composed of. This resulted in the equation 'a' squared plus 'b' squared equals 'c' squared. Proving this gives us confidence that the Pythagorean Theorem is true for ANY right triangle. The Pythagorean Theorem can be used to determine an unknown side of a right triangle given two other sides. Now, if only Sina could help Maurice have a eureka moment for science!