To solve the questions:

4.1 Use the theorem of Pythagoras to calculate the diameter of the circle.

The diagram shows a right triangle inscribed in a semicircle, with the right angle opposite the hypotenuse. The sides of the triangle are given as 5 cm and 12 cm. We can use the Pythagorean theorem to find the hypotenuse, which will be the diameter of the circle.

The Pythagorean theorem states: $c^2 = a^2 + b^2$ where:

• $a = 5 \, \text{cm}$
• $b = 12 \, \text{cm}$

Substituting the values: $c^2 = 5^2 + 12^2$ $c^2 = 25 + 144$ $c^2 = 169$ $c = \sqrt{169} = 13 \, \text{cm}$

Thus, the diameter of the circle is 13 cm.

4.2 Calculate the area of the shaded area in the diagram, correct to two decimal places.

To find the shaded area, we need to subtract the area of the triangle from the area of the semicircle.

1. Area of the semicircle:

   • The diameter is 13 cm, so the radius $r$ is: $r = \frac{13}{2} = 6.5 \, \text{cm}$
   • The area of a full circle is $\pi r^2$, so the area of the semicircle is: $\text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (6.5)^2$ $= \frac{1}{2} \pi \times 42.25$ $\approx 66.48 \, \text{cm}^2$

2. Area of the triangle:

   • The area of a right triangle is given by: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 12$ $= \frac{1}{2} \times 60 = 30 \, \text{cm}^2$

3. Shaded area:

   • Subtract the area of the triangle from the area of the semicircle: $\text{Shaded area} = 66.48 - 30 = 36.48 \, \text{cm}^2$

Thus, the area of the shaded region is approximately 36.48 cm².

Would you like more detailed steps or have any questions?

Related Questions

1. How would the calculations change if the diameter was doubled?
2. What if the triangle was not a right triangle? How would that affect the solution?
3. Can we calculate the circumference of the semicircle based on the diameter?
4. How is the Pythagorean theorem used to find the distance in other geometric shapes?
5. What are other methods to calculate the area of irregular shapes?

Tip:

Remember, in a right triangle inscribed in a circle, the hypotenuse always serves as the diameter of the circle.