Use the image to answer the question.
A figure shows a square made from four right-angled triangles that all have the same dimensions. Each of the four right angled triangles have a height labeled a, a base labeled b, and a hypotenuse labeled c. Sides a and b are positioned so that the right angle creates the four outer corners of the outer square. Each vertex of the inner square divides each side of the outer square in two unequal parts labeled a and b, where a is the shorter part and b is the longer part. Each side of the inner square, labeled c, becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square. The four right-angled triangles are shaded.
Explain a proof of the Pythagorean Theorem using the image.
(4 points)
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
In the image provided, we have a square made from four right-angled triangles of the same dimensions. Each right-angled triangle has sides labeled a, b, and c. The sides a and b are positioned to form the outer square, while the side c becomes the hypotenuse of the four right-angled triangles formed at the corners of the outer square.
Now, let's consider one of these right-angled triangles. We can see that it is formed by two unequal parts of side a (the shorter part) and side b (the longer part), with the hypotenuse being side c.
According to the Pythagorean Theorem, the square of the length of the hypotenuse (c^2) is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2).
In the image, we can see that the four right-angled triangles form a square, with each side of the square being the hypotenuse (c) of one of the triangles. Therefore, the area of the inner square is equal to the sum of the areas of the four right-angled triangles.
The area of the inner square can be calculated by multiplying the lengths of its sides together, which is (a + b)^2.
On the other hand, the area of the inner square can also be calculated by adding up the areas of the four right-angled triangles, which is 4 × (1/2 × a × b).
Setting these two expressions for the area of the inner square equal to each other, we have:
(a + b)^2 = 4 × (1/2 × a × b)
Expanding the square on the left side, we get:
a^2 + 2ab + b^2 = 2ab
Now, subtracting 2ab from both sides:
a^2 + b^2 = 0
This is the Pythagorean Theorem, where the square of the length of the hypotenuse (c^2) is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2).
Therefore, the image provides a proof of the Pythagorean Theorem.
To prove the Pythagorean Theorem using the given image, we can consider the squares and right-angled triangles shown in the figure.
Step 1: Start by observing the larger square, which is formed by combining the four right-angled triangles. Each side of the square measures (a+b), where 'a' is the shorter part and 'b' is the longer part of the division created by the vertices of the inner square.
Step 2: Notice that each side of the inner square measures c, which is also the hypotenuse of the four right-angled triangles formed at the corners of the outer square.
Step 3: Now, examine one of the right-angled triangles shaded in the figure. By the Pythagorean Theorem, we know that the sum of the squares of the lengths of the two shorter sides (a^2 + b^2) is equal to the square of the length of the hypotenuse (c^2).
Step 4: Since there are four right-angled triangles in total, we can write the equation as follows:
4(a^2 + b^2) = (a+b)^2
Expanding the equation, we get:
4a^2 + 4b^2 = a^2 + 2ab + b^2
Step 5: Simplifying the equation, we have:
3a^2 + 3b^2 = 2ab
Step 6: This equation states that the sum of the squares of the two shorter sides is equal to twice the product of the two shorter sides. This proves the Pythagorean Theorem.
Thus, using the image as a visual aid, we have shown a step-by-step proof of the Pythagorean Theorem.
To prove the Pythagorean Theorem using the given image, we can follow these steps:
Step 1: Identify the lengths and relationships in the figure.
- The outer square has side length (a + b).
- Each right-angled triangle has height a, base b, and hypotenuse c.
- The inner square has side length c.
Step 2: Observe the area of the outer square.
- The area of a square is given by the formula A = side^2.
- In this case, the area of the outer square is (a + b)^2.
Step 3: Calculate the area of the outer square using the given figure.
- Divide the outer square into smaller shapes: the four right-angled triangles and the inner square.
- The area of the outer square is equal to the combined area of these smaller shapes.
- The area of each right-angled triangle is (1/2)*a*b. Since there are four such triangles, the total area contributed by them is 4*(1/2)*a*b = 2ab.
- The area of the inner square is c^2.
Step 4: Express the area of the outer square in terms of a, b, and c.
- Combining the above calculations, the total area of the outer square is (a + b)^2 = 2ab + c^2.
Step 5: Set the two expressions for the area of the outer square equal to each other.
- Equate the formulas for the outer square's area:
(a + b)^2 = 2ab + c^2
Step 6: Simplify the equation.
- Expand (a + b)^2: a^2 + 2ab + b^2 = 2ab + c^2
- Simplify: a^2 + b^2 = c^2
Step 7: Recognize the result.
- The final equation obtained is a^2 + b^2 = c^2, which is the Pythagorean Theorem.
Therefore, using the given image, we have proved the Pythagorean Theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).