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.
[tex]a=60^{\circ}[/tex]
[tex]x=3 in[/tex].
The perimeter of the rhombus is 16 inches.
The measure of the greater interior angle of the rhombus is [tex]90^{\circ}[/tex].
The length of the longer diagonal is approximately 7
Answer (2)
The acute angle a of the rhombus is 6 0 ∘ .
The side length s of the rhombus is calculated from the perimeter: s = 4 16 = 4 inches.
The length of the longer diagonal is 4 3 ≈ 6.928 inches, which is approximately 7 inches.
The three true statements are identified: a = 6 0 ∘ , perimeter is 16 inches, and the longer diagonal is approximately 7 inches.
a = 6 0 ∘ , Perimeter = 16 inches , Longer diagonal ≈ 7 inches
Explanation
Problem Analysis Let's analyze the given problem. We have a rhombus formed by four congruent triangles, and one of its diagonals is equal to the side length. This special condition allows us to determine the angles and other properties of the rhombus.
Finding the acute angle Let s be the side length of the rhombus. Since one of the diagonals is equal to the side length, this diagonal divides the rhombus into two equilateral triangles. Therefore, the acute angle of the rhombus, a , is 6 0 ∘ . So, the statement a = 6 0 ∘ is true.
Finding the side length The perimeter of the rhombus is given as 16 inches. Since a rhombus has four equal sides, we have 4 s = 16 , which means s = 4 inches.
Finding the longer diagonal The longer diagonal, d 2 , can be found using the law of cosines or by recognizing that the longer diagonal divides the rhombus into two congruent triangles with sides s , s , and d 2 , where the angle opposite d 2 is 12 0 ∘ (since the angles of a rhombus are supplementary, the obtuse angle is 18 0 ∘ − 6 0 ∘ = 12 0 ∘ ). Using the law of cosines:
d 2 2 = s 2 + s 2 − 2 s 2 cos ( 12 0 ∘ )
Since cos ( 12 0 ∘ ) = − 2 1 , we have
d 2 2 = 2 s 2 − 2 s 2 ( − 2 1 ) = 2 s 2 + s 2 = 3 s 2
So, d 2 = s 3 . Since s = 4 , d 2 = 4 3 ≈ 4 ( 1.732 ) = 6.928 inches. Therefore, the statement "The length of the longer diagonal is approximately 7 inches" is true.
Finding x The variable x represents half the length of the shorter diagonal. Since the shorter diagonal is equal to the side length s , we have x = 2 s = 2 4 = 2 inches. So, the statement x = 3 in is false.
Finding the obtuse angle The greater interior angle of the rhombus is 12 0 ∘ , not 9 0 ∘ . Therefore, the statement "The measure of the greater interior angle of the rhombus is 9 0 ∘ " is false.
Conclusion Therefore, the three true statements are:
a = 6 0 ∘
The perimeter of the rhombus is 16 inches.
The length of the longer diagonal is approximately 7 inches.
Examples
Rhombuses are commonly found in tile patterns and geometric designs. Understanding their properties, such as the relationship between side lengths, diagonals, and angles, is crucial in architecture and design for creating aesthetically pleasing and structurally sound patterns. For example, knowing the angles and side lengths helps in cutting tiles accurately to fit a specific rhombus-based design.
The three true statements regarding the quilt piece are that the acute angle a is 6 0 ∘ , the perimeter is 16 inches, and the length of the longer diagonal is approximately 7 inches.
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