Question

Fill in the blank with the appropriate article, a, an, or the, or leave the space blank if no article is needed. 1. I want ____ apple from that basket. 2. ____ church on the corner is progressive. 3. Miss Lin speaks ____ Chinese. 4. I borrowed ___ pencil from your pile of pencils and pens. 5. One of the students said, "____ professor is late today." 6 Eli likes to play ____ volleyball. 7. I bought ____ umbrella to go out in the rain. 8. My daughter is learning to play ____ violin at her school. 9. Please give me ____ cake that is on the counter. 10. I lived on ____ Main Street when I first came to town. 11. Albany is the capital of ____ New York State. 12. My husband's family speaks ____ Polish. 13. ____ apple a day keeps the doctor away. 14. ____ ink in my pen is red. 15. Our neighbors have ____ cat and ____ dog.​

Answer 1

The converse of the Pythagorean theorem.

Triangle must satisfy $\sf{\overline{AB}^2=\overline{BC}^2+\overline{CA}^2}$ to become a right triangle.

One of the given condition is $\sf{\overline{BC}=\overline{CA}}$

We get $\sf{\overline{BC}^2=\overline{CA}^2}$

Another given condition is $\sf{\overline{AB}^2=2\overline{CA}^2}$

$\sf{\implies \sf{\overline{AB}^2=\overline{CA}^2+\overline{CA}^2}}$

$\sf{\implies \overline{AB}^2=\overline{BC}^2+\overline{CA}^2}$ (Right-angled at $\sf{\angle{C}}$)

So we have a right triangle according to the converse of the Pythagorean theorem.

Hence proven.

Interesting facts

$\sf{\overline{AB}}$ is √2 times the other sides.

$\sf{\implies\overline{AB}=\sqrt{2} \;\overline{BC}}$

$\sf{\implies\overline{AB}=\sqrt{2} \;\overline{CA}}$

The triangle area is $\dfrac{1}{4}$ times the square of the hypotenuse.

Pythagorean theorem was not proven by himself. It was Euclid who proven it for the first time.

Fermat's last theorem is similar to $\sf{a^2+b^2=c^2}$, but any integer satisfy over the power of 2. One of the interesting facts is Fermat didn't prove it. 358 years since Andrew Wiles came up with the perfect proof.