Given a triangle with sides in the ratio $20 : 21 : 29$ inscribed in a circle with a radius of 5 units, we are tasked with finding the area of the triangle.

Step 1: Identifying the sides

Since the sides are in the ratio $20 : 21 : 29$, we can express the sides as:

• $a = 20k$
• $b = 21k$
• $c = 29k$

The side $c = 29k$ is the longest, and it corresponds to the diameter of the circle because the triangle is inscribed in a circle. The relationship between the diameter and the circumradius $R$ is given by: $c = 2R$ Since the radius $R = 5$, we have: $c = 2 \times 5 = 10$ Thus, $29k = 10$, which gives $k = \frac{10}{29}$.

Step 2: Calculating the sides

Now, substitute $k = \frac{10}{29}$ into the expressions for the sides: $a = 20k = 20 \times \frac{10}{29} = \frac{200}{29}$ $b = 21k = 21 \times \frac{10}{29} = \frac{210}{29}$ $c = 29k = 10$

Step 3: Using the area formula for a triangle inscribed in a circle

The formula for the area of a triangle inscribed in a circle is: $\text{Area} = \frac{abc}{4R}$ Substituting the values of $a$, $b$, $c$, and $R$: $\text{Area} = \frac{\frac{200}{29} \times \frac{210}{29} \times 10}{4 \times 5}$ Simplifying: $\text{Area} = \frac{\frac{200 \times 210 \times 10}{29^2}}{20}$ $\text{Area} = \frac{200 \times 210 \times 10}{20 \times 29^2}$ $\text{Area} = \frac{420000}{29^2 \times 20} = \frac{420000}{16820} \approx 24.97 \, \text{sq. units}$

Final Answer:

The area of the triangle is approximately $25 \, \text{sq. units}$.

Would you like more details on any of these steps?

Here are five related questions you might be interested in:

1. How do you find the circumradius of a triangle using other properties?
2. What is the significance of the inscribed circle in a triangle?
3. How can Heron's formula be applied to find the area of a triangle?
4. What are other methods for calculating the area of a triangle inscribed in a circle?
5. Can this method be extended to other polygons?

Tip: When a triangle is inscribed in a circle, and one side equals the diameter, the triangle is always a right triangle.