To find the value of $y$ in the right-angled triangle, we can use the Pythagorean theorem, which states:

$a^2 + b^2 = c^2$

where $a$ and $b$ are the legs (the shorter sides) of the triangle, and $c$ is the hypotenuse (the longest side, opposite the right angle).

In this triangle:

• One leg is 3 cm.
• The other leg is 7 cm.
• The hypotenuse is $y$.

So we can apply the theorem as follows:

$3^2 + 7^2 = y^2$

Calculating this:

$9 + 49 = y^2$

$y^2 = 58$

$y = \sqrt{58}$

Let me calculate $\sqrt{58}$ and round it to 1 decimal place.The value of $y$, rounded to one decimal place, is 7.6 cm.

Do you have any questions or would you like further details? Here are five related questions to explore the topic further:

1. How do you apply the Pythagorean theorem to different right-angled triangles?
2. Can the Pythagorean theorem be used for non-right-angled triangles?
3. What is the general formula for finding the length of any side in a right-angled triangle?
4. How would you approach this problem if the value of the hypotenuse were given instead of the two legs?
5. How does the Pythagorean theorem relate to the distance formula in coordinate geometry?

Tip: In a right triangle, the hypotenuse is always the longest side and directly opposite the right angle.