To solve these two problems, let's use the Pythagorean theorem, which applies to right-angled triangles.

The Pythagorean theorem states: $a^2 + b^2 = c^2$ where $c$ is the hypotenuse (the side opposite the right angle), and $a$ and $b$ are the other two sides.

Solution to Part (a)

In this triangle:

• The sides given are $2.3 \, \text{cm}$ and $1.6 \, \text{cm}$.
• We are trying to find $x$, the hypotenuse.

Using the Pythagorean theorem: $x^2 = (2.3)^2 + (1.6)^2$ $x^2 = 5.29 + 2.56$ $x^2 = 7.85$ $x = \sqrt{7.85} \approx 2.8 \, \text{cm}$

So, $x \approx 2.8 \, \text{cm}$.

Solution to Part (b)

In this triangle:

• The sides given are $4 \, \text{cm}$ and $6 \, \text{cm}$.
• We are trying to find $y$, one of the legs, where $6 \, \text{cm}$ is the hypotenuse.

Using the Pythagorean theorem: $y^2 + (4)^2 = (6)^2$ $y^2 + 16 = 36$ $y^2 = 36 - 16$ $y^2 = 20$ $y = \sqrt{20} \approx 4.47 \, \text{cm}$

So, $y \approx 4.47 \, \text{cm}$.

Final Answers:

• (a) $x \approx 2.8 \, \text{cm}$
• (b) $y \approx 4.47 \, \text{cm}$

Would you like more details on any part of the solution, or do you have any questions?

Additional Questions:

1. How does the Pythagorean theorem apply to non-right triangles?
2. What are some real-life applications of the Pythagorean theorem?
3. Can the values of $x$ and $y$ be solved using trigonometric ratios instead of the Pythagorean theorem?
4. How would you solve for an unknown angle in a right triangle with two known side lengths?
5. Why is it essential to identify the hypotenuse when applying the Pythagorean theorem?

Tip:

Always ensure the triangle has a right angle before using the Pythagorean theorem.