Let's analyze the given problem step by step.
1. Given Information:
- One of the diagonals of the rhombus is equal to its side length.
- [tex]\(a = 60^{\circ}\)[/tex]
- [tex]\(x = 3\)[/tex] inches, which is the length of the side of the rhombus.
2. Side Length of the Rhombus:
- The side length of the rhombus is [tex]\(x = 3\)[/tex] inches.
3. Perimeter of the Rhombus:
- The perimeter of a rhombus is calculated by [tex]\(P = 4 \times \text{side length}\)[/tex].
- Therefore, [tex]\(P = 4 \times 3 = 12\)[/tex] inches.
4. Angle within the Rhombus:
- Given that [tex]\(a = 60^{\circ}\)[/tex], [tex]\(a\)[/tex] is the measure of one of the interior angles.
- Since rhombus angles are supplementary in pairs and the congruent triangles inside form a right angle, we know the measure of the greater interior angle is [tex]\(90^{\circ}\)[/tex].
5. Length of the Longer Diagonal:
- To find the longer diagonal, use the property of the right triangles formed within the rhombus.
- Given [tex]\(a = 60^{\circ}\)[/tex], which is commonly associated with special right triangles (30-60-90 triangles).
- Using the relationship from this triangle, the longer diagonal is calculated to be approximately [tex]\( 4.242640687119286 \)[/tex] inches.
Based on the true results obtained, let's verify which of the given options are correct:
a. [tex]\(a=60^{\circ}\)[/tex]: This is true.
b. [tex]\(x=3\)[/tex] in.: This is true.
c. The perimeter of the rhombus is 16 inches: This is false (as the correct perimeter is 12 inches).
d. The measure of the greater interior angle of the rhombus is [tex]\(90^{\circ}\)[/tex]: This is true.
e. The length of the longer diagonal is approximately 7 inches: This is false (as the correct length is approximately [tex]\(4.242640687119286\)[/tex] inches).
Therefore, the three true measures for the quilt piece are:
1. [tex]\(a=60^{\circ}\)[/tex]
2. [tex]\(x=3\)[/tex] in.
3. The measure of the greater interior angle of the rhombus is [tex]\(90^{\circ}\)[/tex].
