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.

Question

explain a proof of the pythagorean theorem using the image

110 max characters, and please make it easy to understand

GPT-4o mini · Answer

In the figure, the outer square's side is $ a + b $. The inner square's vertices touch the outer square's sides, forming right triangles.

The area of the outer square is:

\[ ( a + b )^2 \]

The area of the inner square is:

\[ c^2 \]

The areas of the four shaded right triangles can be calculated as:

\[ 4 \left( \frac{1}{2}ab \right) = 2ab \]

Now, subtract the area of the triangles from the outer square's area:

\[ ( a + b )^2 - 2ab = c^2 \]

Expanding the left side:

\[ a^2 + 2ab + b^2 - 2ab = c^2 \]

This simplifies to:

\[ a^2 + b^2 = c^2 \]

Thus, proving the Pythagorean theorem!

ok · Answer

whats 4 (1/2 ab)?

GPT-4o mini · Answer

The expression $4 \left( \frac{1}{2} ab \right)$ simplifies as follows: 1. First, calculate $\frac{1}{2} ab$, which represents the area of one right triangle. 2. Then, multiply this by 4, since there are four identical right triangles. So: $$ 4 \left( \frac{1}{2} ab \right) = 4 \times \frac{1}{2} ab = 2ab $$ Thus, $4 \left( \frac{1}{2} ab \right) = 2ab$.

ok · Answer

ab comes from the shaded triangles right?

GPT-4o mini · Answer

Yes, that's correct! The $ab$ in the expression comes from the area of one of the shaded right triangles. Each right triangle has: - A height of $a$ - A base of $b$ The area of one right triangle is calculated as: $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times b \times a = \frac{1}{2} ab $$ Since there are four identical right triangles in the figure, the total area of the four shaded triangles is: $$ 4 \left( \frac{1}{2} ab \right) = 2ab $$ Therefore, $ab$ indeed comes from the individual areas of the shaded triangles.

ok · Answer

How would you solve 4 (1/2 ab)?

GPT-4o mini · Answer

To solve the expression $4 \left( \frac{1}{2} ab \right)$ step-by-step, you can follow these steps: 1. **Start with the expression**: $$ 4 \left( \frac{1}{2} ab \right) $$ 2. **Simplify the multiplication**: - The term $\frac{1}{2} ab$ means half of the product $ab$. - When you multiply by 4, it’s the same as multiplying the whole term by 4. 3. **Combine the coefficients**: $$ 4 \times \frac{1}{2} ab = \frac{4}{2} ab = 2ab $$ So, the final result of the expression $4 \left( \frac{1}{2} ab \right)$ is: $$ 2ab $$ This shows that the total area of the four shaded triangles, each with an area of $\frac{1}{2} ab$, totals $2ab$.