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. $a = 60^{circ}$ $x = 3$ in. the perimeter of the rhombus is 16 inches. the measure of the greater interior angle of the rhombus is $90^{circ}$. the length of the longer diagonal is approximately 7 inches.

Explanation:

Step1: Check the value of \(a\)

Since the triangle has a \(30^{\circ}\) angle and the side - length of the rhombus (hypotenuse of the right - triangle) is \(4\) inches and one of the diagonals (shorter diagonal) is \(4\) inches. In a right - triangle, if the hypotenuse is \(4\) inches and one of the legs (shorter diagonal half) is \(2\) inches (\(\sin30^{\circ}=\frac{2}{4}\)), the other acute angle \(a = 60^{\circ}\) (because the sum of angles in a triangle is \(180^{\circ}\), \(180-(90 + 30)=60\)).

Step2: Calculate the perimeter of the rhombus

The side - length of the rhombus \(s = 4\) inches. The perimeter of a rhombus \(P=4s\). Substitute \(s = 4\) into the formula: \(P = 4\times4=16\) inches.

Step3: Calculate the length of the longer diagonal

Using the Pythagorean theorem in the right - triangle (half of the longer diagonal \(y\)): \(y=\sqrt{4^{2}-2^{2}}=\sqrt{16 - 4}=\sqrt{12}=2\sqrt{3}\approx3.46\) inches. The length of the longer diagonal \(d = 2y\approx2\times3.46\approx7\) inches.

Step4: Check the value of \(x\)

Using the Pythagorean theorem \(x=\sqrt{4^{2}-2^{2}}=\sqrt{12}\approx3.46
eq3\)

Step5: Check the measure of the interior angles

The adjacent - angles of a rhombus are supplementary. If one angle is \(60^{\circ}\), the other is \(120^{\circ}
eq90^{\circ}\)

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.