The diagram shows a right triangle, where we are given two sides:

• $AB = 9$
• $AC = 6$

To find the distance $BC$, we can use the Pythagorean theorem, which applies to right-angled triangles:

$AB^2 = AC^2 + BC^2$

Rearranging this equation to solve for $BC$:

$BC^2 = AB^2 - AC^2$

Now let's substitute the known values:

$BC^2 = 9^2 - 6^2$ $BC^2 = 81 - 36 = 45$

Therefore:

$BC = \sqrt{45} = 3\sqrt{5} \approx 6.71$

So, the distance $BC$ is approximately 6.71 units.

Would you like further details or have any questions? Here are five related questions to expand on this:

1. How would the Pythagorean theorem change if the triangle was not a right triangle?
2. Can you calculate the area of the triangle with the known side lengths?
3. What would happen if $AC$ and $BC$ were swapped? How would the calculation differ?
4. Can you explore different methods for calculating the distance between points in non-right triangles?
5. How can trigonometric ratios be used to find unknown sides in right triangles?

Tip: When using the Pythagorean theorem, always ensure you know which side is the hypotenuse (the longest side opposite the right angle).