Explain a proof of the Pythagorean Theorem using the image. The image is the image of four right triangles and a square proof of the Pythagorean Theorem.
make it how a 7th grader would write it.
The Pythagorean Theorem is a very useful math rule that helps us find the length of a missing side of a right triangle. Let's understand it using a cool image!
Look at this picture. It shows a square shape divided into four smaller right triangles. Each of these triangles has a right angle (90 degrees) and two shorter sides. One side is horizontal, and the other is vertical.
Take a close look at the triangles. Notice that the shortest side of a triangle is actually a part of the square's side. And the longest side of a triangle is the diagonal of the square.
Now, let's examine the relationship between the sides of the triangle. According to the Pythagorean Theorem, the square of the longest side (the hypotenuse) is equal to the sum of the squares of the other two sides (the legs).
For example, if the shortest side (one leg) has a length of 3 units, and the other leg is 4 units, we can use the Pythagorean Theorem to find the hypotenuse. We square the lengths of both legs: 3^2 = 9 and 4^2 = 16. Then, we add those two results together: 9 + 16 = 25.
Now, we need to find the square root of 25 to get the hypotenuse's length. The square root of 25 is 5.
So, using the Pythagorean Theorem, we found that the hypotenuse (the longest side) of this triangle is 5 units. This rule works every time for any right triangle!
It's like magic! We can find any missing side of a right triangle by using the Pythagorean Theorem. Just remember the formula: a² + b² = c², where 'a' and 'b' are the two legs, and 'c' is the hypotenuse.
shorter pls
Sure!
The Pythagorean Theorem says that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In the picture, we see a square divided into four smaller right triangles. Each triangle has one side that is part of the square's side, and another side that is part of the square's diagonal.
If we know the lengths of the two shorter sides of a right triangle, we can use the Pythagorean Theorem to find the length of the hypotenuse.
For example, if one side is 3 units long and the other side is 4 units long, we square both lengths: 3² = 9 and 4² = 16. Then, we add those two results: 9 + 16 = 25.
To find the length of the hypotenuse, we need to take the square root of 25, which is 5.
So, using the Pythagorean Theorem, we found that the hypotenuse of this triangle is 5 units. This rule helps us find missing sides of right triangles. It's like magic! Remember the formula: a² + b² = c², where a and b are the two shorter sides, and c is the hypotenuse.
sumerize this
Sure! Let's explain the Pythagorean theorem using the image.
To start with, what is the Pythagorean theorem? It's a super important math rule that says in a right triangle (a triangle with one angle measuring 90 degrees), the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. In simple words, it helps us find the length of one side of a right triangle when we know the lengths of the other two sides.
Now, let's look at the image. Imagine we have a square with sides labeled "a" and "b". Inside this square, we can see four right triangles, each with one side along the edge of the square. The hypotenuse of each triangle is a side of the square.
Now, how can we show that the Pythagorean theorem works using this image? Well, let's start by finding the area of the entire square. The area of a square is found by multiplying its side length by itself. So the area of our square is (a + b) multiplied by (a + b), which can be written as (a + b)².
Next, let's find the areas of the individual four right triangles. The area of a triangle is found by multiplying one-half of the base by the height. The base of each triangle is either "a" or "b", and the height is also either "a" or "b". So the areas of the four triangles are: 1/2 * a * a, 1/2 * b * b, 1/2 * a * b, and 1/2 * b * a.
Now, let's add up the areas of the four triangles. We get: 1/2 * a * a + 1/2 * b * b + 1/2 * a * b + 1/2 * b * a. Simplifying this, we have: 1/2 * a² + 1/2 * b² + a * b + b * a. Now, notice that 1/2 * a² + 1/2 * b² is the same as (1/2) * (a² + b²). Also, a * b + b * a is the same as 2ab.
So, the total area of the four triangles is (1/2) * (a² + b²) + 2ab, which can also be written as (a² + b²) + 2ab.
Finally, we compare this to the area of the square, which is (a + b)². According to the Pythagorean theorem, these two areas must be equal. Meaning, we can write an equation: (a + b)² = (a² + b²) + 2ab.
Now, let's simplify this equation. When we square (a + b), we get (a + b) * (a + b), which simplifies to a² + 2ab + b². Our equation now becomes a² + 2ab + b² = a² + b² + 2ab.
If we subtract a² and b² from both sides of the equation, we get 2ab = 2ab. And look, both sides are equal! That means our equation is true.
To summarize, our equation (a + b)² = a² + b² + 2ab shows that the area of the square is the same as the sum of the four triangles' areas. And that's the Pythagorean theorem!
So, next time when you need to find the length of one side of a right triangle, use this theorem. Just plug in the lengths of the other two sides, and you'll be able to solve it.