Answer: 26.18 cm
Explanation:
To calculate the length of the arc HI, you need to find the radius of the circle (since the center of the circle is the midpoint of FG) and then use the formula for the length of an arc.
Given:
FG = 80 cm
HF = 136 cm
The radius of the circle is the average of FG and HF because the center is the midpoint of FG:
Radius (r) = (FG + HF) / 2
Radius (r) = (80 cm + 136 cm) / 2
Radius (r) = 216 cm / 2
Radius (r) = 108 cm
Now that we have the radius, you can calculate the length of the arc HI using the formula:
Arc Length (L) = (θ/360) * (2πr)
First, you need to find the angle θ at the center of the circle formed by arc HI. To do this, you can use the properties of similar triangles since FH is a chord of the circle and HI is an arc of the same circle.
You have:
HF (chord length) = 136 cm
FG (chord length) = 80 cm
Let's call θ the angle at the center of the circle (the angle HI makes at the center of the circle). Using similar triangles:
(sin(θ/2)) = (opposite side) / (hypotenuse)
(sin(θ/2)) = (80 cm / 2) / 108 cm
(sin(θ/2)) = 40 cm / 108 cm
(sin(θ/2)) ≈ 0.3704
Now, you can find θ by taking the arcsin of 0.3704:
θ/2 ≈ arcsin(0.3704)
θ/2 ≈ 21.84 degrees
Now, double θ/2 to find θ:
θ ≈ 2 * 21.84 degrees
θ ≈ 43.68 degrees
Now that you have θ and the radius (r), you can calculate the arc length (L):
Arc Length (HI) = (θ/360) * (2πr)
Arc Length (HI) = (43.68/360) * (2π * 108 cm)
Arc Length (HI) ≈ (0.121) * (216π cm)
Arc Length (HI) ≈ 26.18 cm (rounded to two decimal places)
So, the length of the arc HI is approximately 26.18 cm.