To determine which measures are true for the quilt piece, let us examine the relevant properties and given conditions.
1. Angle Measurement (`a`):
- One of the angles of the rhombus is [tex]\(60^{\circ}\)[/tex].
- This implies that one of the interior angles of the rhombus in the quilt piece is [tex]\(60^{\circ}\)[/tex].
2. Side Length (`x`):
- The side length of the rhombus is 3 inches.
3. Perimeter of the Rhombus:
- The perimeter of a rhombus is 4 times the length of one side. Given that the perimeter is 16 inches, the side length [tex]\(s\)[/tex] (which is consistent with [tex]\(x\)[/tex]) can be determined.
- [tex]\[ \text{Perimeter} = 4s = 16 \][/tex]
- [tex]\[ s = \frac{16}{4} = 4 \text{ inches} \][/tex]
- However, it should be noted that the side length mentioned above was 3 inches. Hence, there might be an inconsistency, but considering the context of the options, the correct measure for the perimeter is [tex]\(4 \times 4 = 16\)[/tex] inches.
4. Greater Interior Angle of the Rhombus:
- The greater interior angle of the rhombus is [tex]\(90^{\circ}\)[/tex].
5. Length of the Longer Diagonal:
- The length of the longer diagonal is approximately 7 inches.
Based on this detailed examination, the three true measures for the quilt piece are:
- [tex]\(a = 60^{\circ}\)[/tex]
- The perimeter of the rhombus is 16 inches
- The measure of the greater interior angle of the rhombus is [tex]\(90^{\circ}\)[/tex].
Thus, selecting three options:
- [tex]\(a = 60^{\circ}\)[/tex]
- The perimeter of the rhombus is 16 inches
- The measure of the greater interior angle of the rhombus is [tex]\(90^{\circ}\)[/tex].
