https://file.aiquickdraw.com/mathbot/file/17300935773980f10vfuy.jpg

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.

Find the values of unknown lengths in each of the following triangles (shown in image):
(a) Triangle with sides 2.3 cm and 1.6 cm and hypotenuse labeled x.
(b) Triangle with sides 4 cm and hypotenuse 6 cm, with one leg labeled y.

Learn to find unknown side lengths in right triangles using the Pythagorean theorem. This problem includes calculating hypotenuse and leg lengths for triangles with given side lengths of 2.3 cm, 1.6 cm, 4 cm, and 6 cm. Ideal for Grades 7-9 geometry practice.

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