Question

a quilt piece is designed with four congruent triangles to form a rhombus so that one of the diagonals is equal to the side length of the rhombus. which measures are true for the quilt piece? select three options. a = 60° x = 3 in. the perimeter of the rhombus is 16 inches. the measure of the greater interior angle of the rhombus is 90°. the length of the longer diagonal is approximately 7 inches.

Explanation:

Step1: Analyze angle a

Since one diagonal of the rhombus is equal to the side - length of the rhombus, the two adjacent sides and this diagonal form an equilateral triangle. In an equilateral triangle, each angle is 60°. So, $a = 60^{\circ}$.

Step2: Calculate x

In a right - triangle formed by the diagonals of the rhombus, using the Pythagorean theorem. The side of the rhombus is 4 in, half of the shorter diagonal is 2 in. Let the half - length of the other diagonal be $x$. Then $x=\sqrt{4^{2}-2^{2}}=\sqrt{16 - 4}=\sqrt{12}=2\sqrt{3}\approx3.46$ in, so $x
eq3$ in.

Step3: Calculate the perimeter

The side - length of the rhombus $s = 4$ in. The perimeter of a rhombus $P = 4s$. Substituting $s = 4$ in, we get $P=4\times4 = 16$ inches.

Step4: Analyze the interior angles

The smaller interior angle of the rhombus is 60° (from the equilateral - triangle formed), and the greater interior angle is $180 - 60=120^{\circ}
eq90^{\circ}$.

Step5: Calculate the longer diagonal

The length of the longer diagonal $d_{2}=2x$. Since $x = 2\sqrt{3}\approx3.46$ in, $d_{2}=4\sqrt{3}\approx6.93\approx7$ inches.

Answer:

A. $a = 60^{\circ}$
C. The perimeter of the rhombus is 16 inches.
E. The length of the longer diagonal is approximately 7 inches.