𝑀𝐴𝐵 is a right-angled triangle at 𝑀, with an area of 58 centimeters squared. Find the perimeter of the coloured part of the figure, giving the answer to two decimal places.
So looking at the diagram, we can see that 𝑀𝐴𝐵 is a right-angled triangle, with one vertex 𝑀 at the center of the circle and its other two vertices, 𝐴 and 𝐵, on the circumference of the circle. We’re told that its area is 58 centimeters squared and asked to use this information to find the perimeter of the coloured part of the diagram. This perimeter consists of three parts, two straight lines 𝐴𝑀 and 𝐵𝑀, and then an arc 𝐴𝐵. The two lines 𝐴𝑀 and 𝐵𝑀 are both radii at the circle, as their end points are the center of the circle and a point on the circumference. Therefore, both of these lines could be replaced with 𝑟 in our expression for the perimeter.
Now let’s consider the arc length 𝐴𝐵. This is a portion of the full circumference of the circle. And as the angle at the center is a right angle, it is in fact a quarter of the circumference. The formula for calculating the circumference of the circle is two 𝜋𝑟. And therefore, this arc 𝐴𝐵 will be two 𝜋𝑟 over four. This whole expression for the perimeter can be simplified to give two 𝑟 plus 𝜋𝑟 over two.
So, in order to calculate the perimeter of the coloured part of the figure, we need to know the radius of the circle. Remember, we’ve been told the area of the right-angled triangle. The area of a right-angled triangle is usually calculated using the formula base times height over two. In this question, both the base and the perpendicular height of the triangle are the radius of the circle, 𝑟. Therefore, we can form an equation. 𝑟 multiplied by 𝑟, which is 𝑟 squared, over two is equal to 58. We’ll now solve this equation to find the radius of the circle.
The first step is to multiply both sides of the equation by two. This gives 𝑟 squared is equal to 116. Next, we need to take the square root of both sides. So, we have that 𝑟 is equal to the square root of 116. Now, 116 has a square factor four. It’s equal to four multiplied by 29. So, we can express this square root as the square root of four multiplied by 29, which then simplifies to two root 29.
Now that we know the radius of the circle, we can substitute it into our expression for the perimeter. The perimeter is equal to two multiplied by two root 29 plus 𝜋 multiplied by two root 29 over two. This simplifies to give four root 29 plus 𝜋 root 29. Now, at this stage, we need to use a calculator to evaluate this, as it involves both 𝜋 and a surd, root 29. As a decimal, this is equal to 38.45865.
The question asked us to give the answer to two decimal places. So the final step is to round the answer. And we have that the perimeter of the coloured part of the figure is 38.46. And the units for this are centimeters.