To analyze the quilt piece designed with four congruent triangles forming a rhombus, where one diagonal is equal to the side length of the rhombus, we will evaluate the given options based on the properties of a rhombus.
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Understanding the properties of the rhombus: A rhombus has two diagonals that bisect each other at right angles. If one diagonal (let's call it \( d_1 \)) is equal to the side length (let's call it \( s \)), then we can denote the other diagonal as \( d_2 \). The relationship can be derived from the Pythagorean theorem since the rhombus can be divided into two congruent right triangles by either diagonal.
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Choices Analysis:
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A. \( a = 60^\circ \): This measure might not hold simply because if one diagonal equals the side length and the triangles are congruent, the angles formed would more likely relate to 45 degrees or 90 degrees given the properties of the triangles. The measure of 60 degrees suggests an equilateral triangle scenario which conflicts with the concept of a rhombus here.
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B. \( x = 3 , in. \): If we consider the context of side lengths, this could potentially be true, but we need more information to confirm its validity.
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C. The perimeter of the rhombus is 16 inches.: If each side (s) is equal to 4 inches (assuming option B is true), then the perimeter can be calculated as \( P = 4s \). Hence, \( P = 16 , in.\) could be a true statement.
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D. The measure of the greater interior angle of the rhombus is \( 90^\circ \): If one diagonal equals the side, it is highly likely the resulting triangles form angles that lead to a right angle at the intersection. This measure would imply that the rhombus is a square, confirming this as plausible.
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E. The length of the longer diagonal is approximately 7 inches.: If the diagonal equals the side length, using the properties of right triangles, the longer diagonal can indeed be somewhat less than the maximum derived from the side length dimensions. The exact numerical value would need to be calculated based on how the triangle is laid out.
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Conclusion: Based on the above analysis, the most likely true statements based on the properties of the rhombus and the accompanying geometry would be:
- C: The perimeter of the rhombus is \( 16 , in. \).
- D: The measure of the greater interior angle of the rhombus is \( 90^\circ \).
- B: \( x = 3 , in. \), can be true if side measures align properly.
Thus, the selected three true measures based on reasoning are C, D, and B, assuming B aligns after detailed analysis.
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