The problem presented shows a right triangle $\triangle DEF$, with side $f = 5 \, \text{cm}$, hypotenuse $e = 13 \, \text{cm}$, and asks us to find the length of side $d = EF$.

To solve for $d$, we can use the Pythagorean theorem: $a^2 + b^2 = c^2$ where $c$ is the hypotenuse, and $a$ and $b$ are the other two sides.

In this case:

• $f = 5 \, \text{cm}$ (one of the legs),
• $d = EF$ (the other leg, which we need to find),
• $e = 13 \, \text{cm}$ (the hypotenuse).

Step-by-Step Solution

1. Write down the Pythagorean theorem for $\triangle DEF$: $d^2 + f^2 = e^2$

2. Substitute the given values: $d^2 + 5^2 = 13^2$

3. Calculate $5^2$ and $13^2$: $d^2 + 25 = 169$

4. Isolate $d^2$ by subtracting 25 from both sides: $d^2 = 169 - 25$ $d^2 = 144$

5. Take the square root of both sides to solve for $d$: $d = \sqrt{144}$ $d = 12 \, \text{cm}$

Answer

The length of $EF$ is $\mathbf{12 \, \text{cm}}$.

Would you like further details on this solution, or have any questions?

Here are five related questions that might expand your understanding:

1. How would we solve this if only the angle opposite $f$ was given, instead of both sides?
2. What other properties can we deduce about triangle $DEF$ based on its side lengths?
3. How can we verify if a triangle is a right triangle by using side lengths alone?
4. What if $e$ and $d$ were given, and we needed to find $f$?
5. How does the Pythagorean theorem relate to trigonometric identities?

Tip: Always check the lengths provided in right triangles to see if they follow Pythagorean triples (like 5, 12, 13) for quicker verification.