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)
Math Short Answer Rubric (4 points)
Points Awarded Criteria
4
The answer is clear and correct.
The answer shows that the student used problem-solving skills and reasoning.
All work is shown or explained.
3
The answer is mostly correct. The student made small errors while solving.
The answer shows that the student used some problem-solving skills and reasoning.
Most work is shown or explained.
2
The answer is partially correct. The student made errors while solving.
The answer shows that the student used some problem-solving skills or reasoning.
Some work is shown or explained.
1
The answer is incorrect. The student made errors while solving.
The answer shows little problem-solving skills or reasoning.
Little or no work is shown.
0 The question is not answered.
To prove the Pythagorean Theorem using the image, we can start by considering the two right-angled triangles that share side a. Let's call the length of the hypotenuse of the right-angled triangle on the left side of the image as c1 and the hypotenuse of the right-angled triangle on the right side of the image as c2.
Using the Pythagorean Theorem, we know that for each right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. So, we can write the following equations:
a^2 + b^2 = c1^2 (Equation 1)
a^2 + b^2 = c2^2 (Equation 2)
Now, let's consider the square formed by the four right-angled triangles. The side length of this square is 2a. The diagonal of this square can be found by using c, the hypotenuse of the right-angled triangles formed at the corners of the outer square. The diagonal of the square is equal to the sum of the hypotenuses of the four right-angled triangles, so we can write:
2a = c1 + c2 (Equation 3)
We can substitute c1 and c2 from Equation 1 and Equation 2 into Equation 3:
2a = (a^2 + b^2)^(0.5) + (a^2 + b^2)^(0.5)
Simplifying this equation, we get:
2a = 2(a^2 + b^2)^(0.5)
Dividing both sides by 2, we get:
a = (a^2 + b^2)^(0.5)
Squaring both sides, we get:
a^2 = a^2 + b^2
Simplifying this equation, we get:
0 = b^2
Since b is a length, it cannot be 0. Therefore, this equation is not possible. Thus, we have proven the Pythagorean Theorem, which states that a^2 + b^2 = c^2.
can u make that smaller and in a text
Sure! Here's a smaller, condensed version of the proof:
- We start by considering the two right-angled triangles that share side "a" in the image.
- Using the Pythagorean Theorem, we write the equations:
- "a^2 + b^2 = c1^2" (Equation 1) for the left triangle, and
- "a^2 + b^2 = c2^2" (Equation 2) for the right triangle.
- Next, we consider the square formed by the four right-angled triangles with side length 2a.
- By the properties of the square, the diagonal is equal to the sum of the hypotenuses of the four triangles.
- We write the equation: "2a = c1 + c2" (Equation 3) to express this relationship.
- Substituting c1 and c2 from Equations 1 and 2 into Equation 3, we get "2a = 2(a^2 + b^2)^(0.5)".
- Simplifying the equation, we have "a = (a^2 + b^2)^(0.5)".
- Squaring both sides of the equation, we get "a^2 = a^2 + b^2".
- However, this equation simplifies to "0 = b^2", which is not possible since b is a length.
- Therefore, we've concluded that the Pythagorean Theorem holds: "a^2 + b^2 = c^2".