To determine the correct measures for the quilt piece designed as a rhombus with four congruent triangles, we need to analyze and verify each given option. Here's a detailed, step-by-step solution:
1. Verify [tex]\( a = 60^\circ \)[/tex]:
- The design specifies that the smaller angle, [tex]\( a \)[/tex], in each of the congruent triangles forming the rhombus is [tex]\( 60^\circ \)[/tex].
2. Verify [tex]\( x = 3 \)[/tex] inches:
- The problem statement does not provide explicit clarification regarding what [tex]\( x \)[/tex] represents, but it is stated that [tex]\( x = 3 \)[/tex] inches.
3. Verify the perimeter of the rhombus:
- The perimeter of the rhombus is given as 16 inches.
- Each side of the rhombus, therefore, is [tex]\( \text{side length} = \frac{\text{perimeter}}{4} = \frac{16}{4} = 4 \)[/tex] inches.
4. Measure of the greater interior angle:
- The angles of a rhombus are supplementary, meaning they add up to [tex]\( 180^\circ \)[/tex].
- Since the interior angles already include [tex]\( 60^\circ \)[/tex], the greater angle [tex]\( = 180^\circ - 60^\circ = 120^\circ \)[/tex].
- Hence, the larger interior angle of the rhombus is [tex]\( 120^\circ \)[/tex], not [tex]\( 90^\circ \)[/tex].
5. Length of the longer diagonal:
- To find the length of the longer diagonal, we use geometric properties. The longer diagonal bisects the rhombus into two congruent equilateral triangles. Each triangle has an angle of [tex]\( 60^\circ \)[/tex], and each side of these triangles (which is also the side length of the rhombus) is 4 inches.
- Using trigonometry, the longer diagonal [tex]\( d_1 \)[/tex]:
[tex]\[
d_1 = 2 \times (\text{side length}) \times \cos(30^\circ)
= 2 \times 4 \times \cos(30^\circ)
= 2 \times 4 \times \frac{\sqrt{3}}{2}
= 4\sqrt{3} \approx 6.928 \,\text{inches}.
\][/tex]
- The length of the longer diagonal is therefore approximately [tex]\( 6.928 \)[/tex] inches, close enough to 7 inches.
### Conclusion
From the above analysis, the measures that are true for the quilt piece are:
1. [tex]\( a = 60^\circ \)[/tex]
2. [tex]\( x = 3 \)[/tex] inches
3. The perimeter of the rhombus is 16 inches.
The measure of the greater interior angle of the rhombus actually is [tex]\( 120^\circ \)[/tex], not [tex]\( 90^\circ \)[/tex]. And the length of the longer diagonal is approximately 6.928 inches.
