Question

problems 1–2: here is a quarter-circle. it was created using a circle with a 12-inch diameter. the circle was folded in half and then folded in half again. 1. label the quarter-circle with any important measurements. 2. what is the perimeter of this quarter-circle? 3. here is a shape made of a semicircle and a square. what is the perimeter of this shape?

Explanation:

Problem 1: Label the quarter - circle with important measurements

Step 1: Find the radius

The diameter of the original circle is 12 inches. The radius \(r\) of a circle is half of the diameter, so \(r=\frac{12}{2} = 6\) inches. For the quarter - circle, the two straight sides (the radii of the sector) are each equal to the radius of the original circle, so they are 6 inches long. The curved part is a quarter of the circumference of the original circle.

Step 2: Label the quarter - circle

On the quarter - circle, we can label the two straight sides (the radii) as 6 inches each, and we can note that the curved part is a quarter - arc of a circle with radius 6 inches.

Problem 2: Perimeter of the quarter - circle

Step 1: Recall the formula for the perimeter of a quarter - circle

The perimeter of a quarter - circle \(P\) is composed of two radii (\(2r\)) and a quarter of the circumference of the full circle. The formula for the circumference of a circle is \(C = 2\pi r\) (or \(C=\pi d\), where \(d\) is the diameter). A quarter of the circumference is \(\frac{1}{4}C=\frac{1}{4}\times2\pi r=\frac{\pi r}{2}\) (or \(\frac{1}{4}\times\pi d=\frac{\pi d}{4}\)). So the perimeter of the quarter - circle \(P = 2r+\frac{1}{4}\times2\pi r=2r+\frac{\pi r}{2}\) (or \(P = 2r+\frac{\pi d}{4}\)).

Step 2: Substitute the values

We know that \(d = 12\) inches, so \(r = 6\) inches. Substitute \(r = 6\) into the formula. First, calculate the length of the two radii: \(2r=2\times6 = 12\) inches. Then calculate the length of the quarter - arc: \(\frac{1}{4}\times2\pi r=\frac{1}{4}\times2\pi\times6 = 3\pi\) inches (or using the diameter, \(\frac{\pi d}{4}=\frac{\pi\times12}{4}=3\pi\) inches).

Step 3: Calculate the perimeter

Now, add the lengths of the two radii and the quarter - arc. \(P=12 + 3\pi\). If we use \(\pi\approx3.14\), then \(P=12+3\times3.14=12 + 9.42=21.42\) inches.

Problem 3: Perimeter of the shape made of a semicircle and a square

Step 1: Determine the dimensions

From the grid, we can see that the side length of the square (and also the diameter of the semicircle) is, let's assume the side length of the square (and the diameter of the semicircle) is \(s\). Looking at the grid, if we count the units, the diameter of the semicircle (and the side of the square) is 6 units (since from the bottom to the top of the semicircle, it spans 6 grid units, and the square also has a side length of 6 units).

Step 2: Recall the formula for the perimeter of the composite shape

The perimeter of the shape is composed of three sides of the square and the length of the semicircular arc. The formula for the circumference of a full circle is \(C=\pi d\), so the length of a semicircular arc is \(\frac{1}{2}C=\frac{\pi d}{2}\). The three sides of the square have a total length of \(3s\).

Step 3: Substitute the values

The diameter \(d = 6\) units, and the side length of the square \(s = 6\) units. The length of the semicircular arc is \(\frac{\pi\times6}{2}=3\pi\) units. The length of the three sides of the square is \(3\times6 = 18\) units.

Step 4: Calculate the perimeter

Add the length of the semicircular arc and the length of the three sides of the square. \(P=18 + 3\pi\). If we use \(\pi\approx3.14\), then \(P=18+3\times3.14=18 + 9.42 = 27.42\) units.

Answer:

s:

1. The two straight sides (radii) of the quarter - circle are labeled as 6 inches, and the curved part is a quarter - arc of a circle with radius 6 inches.
2. The perimeter of the quarter - circle is \(12 + 3\pi\approx21.42\) inches (or \(21.42\) inches when \(\pi\approx3.14\)).
3. The perimeter of the composite shape is \(18+3\pi\approx27.42\) units (or \(27.42\) units when \(\pi\approx3.14\)).