Question

5. Find the area of the largest equilateral triangle that can be inscribed in a circle whose diameter is 20cm.

Answer 1

To find the area of the largest equilateral triangle inscribed in a circle whose diameter is 20 cm, we need to calculate the side length of the triangle and use the formula for the area of an equilateral triangle.

Step-by-step explanation:

To find the area of the largest equilateral triangle inscribed in a circle, we need to consider the side length of the triangle. In an equilateral triangle, each angle is 60 degrees, and the sum of all angles in a triangle is 180 degrees. So, each angle in the equilateral triangle is 180 divided by 3, which is 60 degrees. Since the triangle is inscribed in a circle, the angles formed by connecting the center of the circle to the vertices of the equilateral triangle are right angles. This means that the radius of the circle is also equal to the altitude of the equilateral triangle.

Now, let's calculate the radius of the circle. The diameter of the circle is given as 20 cm, so the radius is half of that, which is 10 cm. Since the radius of the circle is also equal to the altitude of the equilateral triangle, we can use the Pythagorean theorem to find the side length of the triangle. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the side length of the equilateral triangle, the altitude is the height, and the base is half of the side length. Using the Pythagorean theorem, we can solve for the side length:

height² + (side_length/2)² = side_length²

10² + (side_length/2)² = side_length²

100 + (side_length/2)² = side_length²

(side_length/2)² = side_length² - 100

side_length²/4 = side_length² - 100

side_length² - 4(side_length²/4) = 100

3(side_length²/4) = 100

side_length²/4 = 100/3

side_length² = (100/3) * 4 = 400/3

side_length = sqrt(400/3) = (20/3)sqrt(3)

Now that we have the side length of the equilateral triangle, we can use the formula to find its area. The area of an equilateral triangle is given by the formula:

Area = (sqrt(3)/4) * side_length²

Plugging in the value of the side length, we get:

Area = (sqrt(3)/4) * ((20/3)sqrt(3))²

Area = (sqrt(3)/4) * (400/9) * 3

Area = (sqrt(3)/4) * (400/3)

Area = (sqrt(3)/4) * (400/3) = (400/4)sqrt(3)

Area = 100sqrt(3)

Answer 2

Explanation

There is only one equilateral triangle inscribed in a circle whose diameter is 20 cm. Now, the triangle of the largest area inscribed in a circle is equilateral.

Having said that, let's calculate the area of such a triangle. Look at the following picture:

To calculate the area of the (green) triangle, we will follow the following procedure:

0. To find the value of a/2,

,

1. to find the value of a,

,

2. To use Heron's formula for the area of a triangle.

Step 1)

r denotes the radius of the black circle, which is exactly half its diameter. Since the diameter of the black circle is 20 cm, we get that

`r=(20)/(2)cm=10\text{ cm.}`

Then, we have

By the cosine trigonometric relation, we have

`\cos (30)=((a)/(2))/(10)\text{.}`

Then,

`\begin{gathered} \frac{\sqrt[]{3}}{2}=((a)/(2))/(10), \\ 10\cdot\frac{\sqrt[]{3}}{2}=(a)/(2), \\ 5\cdot\sqrt[]{3}=(a)/(2), \\ (a)/(2)=5\cdot\sqrt[]{3}. \end{gathered}`

Step 2)

`\begin{gathered} (a)/(2)=5\cdot\sqrt[]{3}, \\ a=2\cdot5\cdot\sqrt[]{3}, \\ a=10\cdot\sqrt[]{3}\text{.} \end{gathered}`

Step 3)

For our triangle is equilateral, all of its sides have the same length (a). Then, the semi-perimeter of the triangle (s), which is the sum of all lengths divided by 2, is

`s=(a+a+a)/(2)=(3)/(2)\cdot a\text{.}`

Now, let's recall Heron's formula for the area (A) of a triangle:

`A=\sqrt[]{s(s-a)(s-a)(s-a)}\text{.}`

In our particular case, it becomes

`A=\sqrt[]{(3)/(2)a((3)/(2)\cdot a-a)^3}\text{.}`

Simplifying it, we get

`\begin{gathered} A=\sqrt[]{(3)/(2)a((3)/(2)\cdot a-a)^3}, \\ A=\sqrt[]{(3)/(2)a((1)/(2)a)^3}, \\ A=\sqrt[]{(3)/(2)a\cdot(1)/(8)a^3}, \\ A=\sqrt[]{(3)/(16)a^4}, \\ A=\frac{\sqrt[]{3}}{4}a^2. \end{gathered}`

Replacing the value of a in the last expression, we get

`\begin{gathered} A=\frac{\sqrt[]{3}}{4}(10\cdot\sqrt[]{3})^2, \\ A=\frac{\sqrt[]{3}}{4}\cdot100\cdot3, \\ A=25\cdot3\cdot\sqrt[]{3}, \\ A=75\cdot\sqrt[]{3}\text{.} \end{gathered}`Answer

The area of the equilateral triangle inscribed in a circle of diameter 20 cm is

75√3 cm².