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? choose three correct answers. the measure of the greater interior angle of the rhombus is 90°. the perimeter of the rhombus is 16 inches. a = 60° the length of the longer diagonal is approximately 7 inches. x = 3 in.

Explanation:

Step1: Check the perimeter

Since the side length of the rhombus is \(4\) inches and the perimeter of a rhombus \(P = 4s\) (where \(s\) is the side - length), \(P=4\times4 = 16\) inches.

Step2: Analyze the angles

In a rhombus, if one of the angles in the right - triangle formed by the diagonals is \(30^{\circ}\), using the properties of a rhombus (diagonals bisect the angles). The smaller interior angle of the rhombus: if the diagonal bisects an angle, and we know the angle in the right - triangle is \(30^{\circ}\), the smaller interior angle of the rhombus is \(60^{\circ}\) (because the diagonal bisects the angle). The larger interior angle of the rhombus is \(180 - 60=120^{\circ}
eq90^{\circ}\). Also, \(a = 60^{\circ}\) (using the property that the diagonal bisects the angle of the rhombus).

Step3: Calculate the length of the longer diagonal

Using the Pythagorean theorem in the right - triangle formed by the half - diagonals and the side of the rhombus. Let the half - length of the longer diagonal be \(y\). We know that the side of the rhombus \(s = 4\) inches and the half - length of the shorter diagonal \(d_1/2=2\) inches. By the Pythagorean theorem \(y=\sqrt{4^{2}-2^{2}}=\sqrt{16 - 4}=\sqrt{12} = 2\sqrt{3}\) inches. The length of the longer diagonal \(d_2=4\sqrt{3}\approx6.93\approx7\) inches. For \(x\) (half - length of the longer diagonal), using \(\cos30^{\circ}=\frac{x}{4}\), \(x = 4\cos30^{\circ}=4\times\frac{\sqrt{3}}{2}=2\sqrt{3}
eq3\)

Answer:

The perimeter of the rhombus is 16 inches.; \(a = 60^{\circ}\); The length of the longer diagonal is approximately 7 inches.