Question

Question

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)
This table shows the criteria for awarding points and the corresponding point values.
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.


Skip to navigation

Answer 1

To prove the Pythagorean Theorem using the described image with the square and four right-angled triangles, we can follow these steps:

1. Identify Key Elements: As per the image description, we have an outer square formed with four right-angled triangles, where each triangle has legs of length \( a \) (height) and \( b \) (base), and hypotenuse \( c \).

2. Understand the Relationship: The placement of the right-angled triangles at the corners of the outer square implies that each vertex of the inner square reduces the dimensions of the outer square's sides. The dimensions of each side of the outer square is \( a + b \), and each triangle's hypotenuse contributes to the sides of the inner square.

3. Area Analysis: The area of the outer square can be calculated as: \[ \text{Area}{\text{outer}} = (a + b)^2 \] Expanding this, we get: \[ \text{Area}{\text{outer}} = a^2 + 2ab + b^2 \]

4. Calculate the Area of Triangles: The area of one right-angled triangle is: \[ \text{Area}{\text{triangle}} = \frac{1}{2}ab \] Since there are four triangles, their total area is: \[ \text{Area}{\text{triangles}} = 4 \times \frac{1}{2}ab = 2ab \]

5. Determine the Area of the Inner Square: The remaining area of the outer square after considering the triangles must be equal to the area of the inner square. Thus: \[ \text{Area}{\text{inner}} = \text{Area}{\text{outer}} - \text{Area}{\text{triangles}} = (a^2 + 2ab + b^2) - 2ab \] This simplifies to: \[ \text{Area}{\text{inner}} = a^2 + b^2 \]

6. Relate to the Hypotenuse: The side length of the inner square is the hypotenuse \( c \) of the right triangles, meaning: \[ \text{Area}_{\text{inner}} = c^2 \]

7. Combine the Equations: Set the areas from the previous steps equal to each other: \[ c^2 = a^2 + b^2 \] This relationship illustrates the Pythagorean theorem, where \( c^2 = a^2 + b^2 \), establishing that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In conclusion, the construction of the outer square, subtraction of the triangles’ areas, and the equivalence of the inner square’s area lead us to prove the Pythagorean Theorem effectively, showcasing that \( c^2 = a^2 + b^2 \) indeed holds true.

This proof is structured, showing clear reasoning and connections between the areas, thus deserving full points.