This image shows a right-angled triangle $PQR$, where $PQ = 8 \, \text{cm}$, $PR = 5 \, \text{cm}$, and we are tasked with finding the length of $QR$ using Pythagoras' theorem.

Pythagoras' Theorem:

For a right-angled triangle, the relationship between the sides is given by:

$QR^2 = PQ^2 + PR^2$

Substituting the values from the diagram:

$QR^2 = 8^2 + 5^2$ $QR^2 = 64 + 25$ $QR^2 = 89$

Now, taking the square root of both sides:

$QR = \sqrt{89} \approx 9.43 \, \text{cm}$

Thus, the length of $QR$ is approximately 9.4 cm (to 1 decimal place).

Would you like more details or have any questions? Here are some additional related questions for practice:

1. What would the length of $QR$ be if $PQ$ was 7 cm instead of 8 cm?
2. How can you use Pythagoras' theorem to determine if a triangle is right-angled given three sides?
3. What is the perimeter of triangle $PQR$ based on the calculated length of $QR$?
4. If the area of the triangle is required, how would you calculate it using the given sides?
5. Can you apply the Pythagorean theorem in 3D geometry, and if so, how?

Tip: Pythagoras' theorem only applies to right-angled triangles, so always check for a 90-degree angle before applying it.