To determine which measures are true for the quilt piece described, we need to analyze the properties of the rhombus and its components. Here's the step-by-step solution:
1. Calculate the Side Length of the Rhombus:
- The perimeter of the rhombus is given as 16 inches.
- The perimeter of a rhombus is equal to [tex]\(4\)[/tex] times the side length.
- So, let's find the side length of the rhombus ([tex]\(s\)[/tex]):
[tex]\[
4s = 16 \implies s = \frac{16}{4} = 4 \text{ inches}
\][/tex]
2. Verify the Measure of Angle [tex]\(a\)[/tex]:
- Given, [tex]\(a = 60^\circ\)[/tex]. This is a stated measure.
- Since the given measure is stated directly, if it matches the conditions set by the properties of a rhombus, it is correct.
3. Verify the Given Side Length [tex]\(x\)[/tex]:
- Given, [tex]\(x = 3\)[/tex] inches.
- From the previous calculation, the side length of the rhombus is [tex]\(4\)[/tex] inches.
- The given side length [tex]\(x\)[/tex] does not match the calculated side length of the rhombus, hence [tex]\(x = 3\)[/tex] inches is not correct.
4. Perimeter of the Rhombus:
- Given, the perimeter is [tex]\(16\)[/tex] inches.
- This matches our calculated perimeter, so this measure is correct.
5. Verify the Measure of the Greater Interior Angle:
- Given, the greater interior angle of the rhombus is [tex]\(90^\circ\)[/tex].
- A rhombus has angles that add up to [tex]\(360^\circ\)[/tex], with opposite angles being equal. Given one of the diagonals is equal to the side length of the rhombus, it would bisect the rhombus into right-angled triangles, making one of the greater interior angles [tex]\(90^\circ\)[/tex].
- Therefore, this measure is correct.
6. Calculate the Length of the Longer Diagonal:
- One of the diagonals is the side length of the rhombus, which is [tex]\(4\)[/tex] inches.
- We need to calculate the longer diagonal. Using the properties of the rhombus (right triangles inside), we can use the Pythagorean theorem:
[tex]\[
\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = s^2
\][/tex]
- Given [tex]\(d_1 = 4\)[/tex] inches, half of this diagonal is [tex]\(2\)[/tex] inches.
- Substituting, we get:
[tex]\[
2^2 + \left(\frac{d_2}{2}\right)^2 = 4^2
\][/tex]
[tex]\[
4 + \left(\frac{d_2}{2}\right)^2 = 16
\][/tex]
[tex]\[
\left(\frac{d_2}{2}\right)^2 = 12
\][/tex]
[tex]\[
\frac{d_2}{2} = \sqrt{12} \approx 3.464
\][/tex]
- Therefore, the longer diagonal:
[tex]\[
d_2 \approx 2 \times 3.464 = 6.928 \text{ inches}
\][/tex]
- Rounding, this is approximately [tex]\(7\)[/tex] inches, so this measure is correct.
Summary of Correct Measures:
- [tex]\(a = 60^\circ\)[/tex]
- The perimeter of the rhombus is [tex]\(16\)[/tex] inches.
- The measure of the greater interior angle of the rhombus is [tex]\(90^\circ\)[/tex].
- The length of the longer diagonal is approximately [tex]\(7\)[/tex] inches.
Thus, the three true measures are:
1. [tex]\(a = 60^\circ\)[/tex]
2. The perimeter of the rhombus is [tex]\(16\)[/tex] inches.
3. The measure of the greater interior angle of the rhombus is [tex]\(90^\circ\)[/tex].
4. The length of the longer diagonal is approximately [tex]\(7\)[/tex] inches.
Since the question asks to select three options, hence the measures that are true for the quilt piece are:
- [tex]\(a = 60^\circ\)[/tex],
- The perimeter of the rhombus is [tex]\(16\)[/tex] inches,
- The measure of the greater interior angle of the rhombus is [tex]\(90^\circ\)[/tex].
