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.

A rhombus with diagonals is shown. The diagonals form 4 triangles. All sides are 4 inches long. The distance from the top point to the middle point is 2 inches. The distance from the right point to the middle point is x. One interior angle is 30 degrees and another is a degrees.

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.

Answer 1

Correct choices: a = 60°, The perimeter of the rhombus is 16 inches, The length of the longer diagonal is approximately 7 inches.

Work:
- Let half-diagonals be p = 2 (given) and q = x. For a rhombus the diagonals are perpendicular and half-diagonals and a side form a right triangle, so 4^2 = 2^2 + x^2 ⇒ x = √12 = 2√3 ≈ 3.464 in (so x = 3 in is false).
- Perimeter = 4·4 = 16 in (true).
- The longer diagonal = 2x = 4√3 ≈ 6.93 ≈ 7 in (true).
- The diagonal bisects the vertex angle; cos(half-angle) = p/side = 2/4 = 1/2 ⇒ half-angle = 60°, so the two rhombus angles are 60° and 120°. Thus a = 60° (true) and the greater interior angle is 120°, not 90° (false).