The images you provided contain three right-angled triangles where you need to find the missing sides. We can solve these using the Pythagorean theorem, which applies to right-angled triangles.

a. First Triangle (4m and 5m):

In this triangle, the two known sides are 4m and 5m. The missing side is the hypotenuse.

The Pythagorean theorem states that:

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

Where:

• $a$ and $b$ are the legs of the triangle,
• $c$ is the hypotenuse.

For this triangle, if 4m and 5m are the legs, we calculate the hypotenuse:

$4^2 + 5^2 = c^2$

$16 + 25 = c^2$

$41 = c^2$

$c = \sqrt{41} \approx 6.4m$

So, the hypotenuse is approximately 6.4 meters.

b. Second Triangle (5cm and 6cm):

Here, we have a right-angled triangle with one leg being 5cm and the other leg being 6cm. We need to find the hypotenuse.

Using the Pythagorean theorem again:

$5^2 + 6^2 = c^2$

$25 + 36 = c^2$

$61 = c^2$

$c = \sqrt{61} \approx 7.81cm$

So, the hypotenuse is approximately 7.81 cm.

c. Third Triangle (2cm and 3cm):

For the third triangle, we are given the legs of 2cm and 3cm, and we need to find the hypotenuse.

Again, applying the Pythagorean theorem:

$2^2 + 3^2 = c^2$

$4 + 9 = c^2$

$13 = c^2$

$c = \sqrt{13} \approx 3.61cm$

So, the hypotenuse is approximately 3.61 cm.

---

Would you like further details or clarifications on any of these steps? Here are some related questions for practice:

1. How can you apply the Pythagorean theorem to find missing legs in a right triangle if the hypotenuse is known?
2. Can the Pythagorean theorem be used for non-right triangles?
3. How do you know if a triangle is a right triangle based on its side lengths?
4. What is the difference between the hypotenuse and the legs of a right triangle?
5. How does the Pythagorean theorem relate to distance calculation in geometry?

Tip: Always double-check the units in your problem. Ensure they are consistent when applying formulas.