The image shows three right-angled triangles, and we are tasked with finding the missing side $x$ in each case using the Pythagorean theorem, which states:

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

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

Left Triangle:

• Known sides: 8 cm and 4 cm.
• To find $x$, the hypotenuse: $x^2 = 8^2 + 4^2$ $x^2 = 64 + 16 = 80$ $x = \sqrt{80} \approx 8.94 \, \text{cm}$

Middle Triangle:

• Known sides: 3 cm and 7.9 cm.
• To find $x$, one of the legs: $x^2 + 3^2 = 7.9^2$ $x^2 + 9 = 62.41$ $x^2 = 62.41 - 9 = 53.41$ $x = \sqrt{53.41} \approx 7.31 \, \text{cm}$

Right Triangle:

• Known sides: 8 cm and 9.2 cm.
• To find $x$, the other leg: $x^2 + 8^2 = 9.2^2$ $x^2 + 64 = 84.64$ $x^2 = 84.64 - 64 = 20.64$ $x = \sqrt{20.64} \approx 4.54 \, \text{cm}$

Final Answers:

1. Left triangle: $x \approx 8.94 \, \text{cm}$
2. Middle triangle: $x \approx 7.31 \, \text{cm}$
3. Right triangle: $x \approx 4.54 \, \text{cm}$

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How does the Pythagorean theorem apply to non-right triangles?
2. What is the geometric significance of the square root in the Pythagorean theorem?
3. Can the Pythagorean theorem be used to determine the area of a triangle?
4. How do you determine if three sides form a right triangle?
5. What are real-life applications of the Pythagorean theorem?

Tip: Always ensure the triangle is a right triangle before applying the Pythagorean theorem!