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? Check all that apply.
1.a = 60°
2.x = 3 in.
3.The perimeter of the rhombus is 16 inches.
4.The measure of the greater interior angle of the rhombus is 90°.
5.The length of the longer diagonal is approximately 7 inches.

Answer 1

Options 1, 3 and 5 are correct.

Explanation:

Option 1.

In the triangle ABO,

30° + 90° + ∠ABO = 180° [ Since diagonals of rhombus intersect at 90°]

∠ABO = 180 - 120 = 60°

Therefore, m∠a = 60° [Since opposite angles of rhombus are always equal]

Option 1 is correct.

Option 2.

Since AO = OC = x [Diagonals of trapezoid bisect each other]

In the triangle AOB,

x² + 2² = 4² [By Pythagoras theorem]

[By the property of trapezoid diagonal intersect each other at 90°]

x² = 16 - 4

x = √12

x = 2√3

Therefore, option 2 is incorrect.

3. Perimeter of the rhombus = 4 + 4 + 4 + 4 = 12

Option 3 is correct.

4. Measure of greater interior angle is = 2 × 60° = 120°

Therefore, option 4 is incorrect.

5. Longer diagonal of the rhombus = 2

= 2 × (2√3)

= 4√3

= 6.92

≈ 7 inches

Therefore, option 5 is correct .

Answer 2

True: 1, 3, 5

Explanation:

The diagram shows the rhombus with the side length of 4 inches.

Properties:

• opposite angles of the rhombus are congruent;
• diagonals of the rhombus bisect each other;
• diagonals of the rhombus bisect rhombus angles;
• diagonals are perpendicular;
• diagonals divide the rhombus into four congruent right triangles with legs that are halves of diagonals.

Consider right triangle (bottom left right triangle) with the hypotenuse of 4 in and acute angle 30°. In this triangle, leg opposite to the 30° angle is half of hypotenuse and is 2 in. By the Pythagorean theorem,

`x^2+2^2=4^2\\ \\x^2=16-4\\ \\x^2=12\\ \\x=√(12)=2√(3)\ in`

So, option 2 is false.

The second acute angle of this triangle has the measure

`a^(\circ)=90^(\circ)-30^(\circ)=60^(\circ)`

So, option 1 is true.

The perimeter of the rhombus is the sum of all sides, so

`P=4+4+4+4=16\ in`

Option 3 is true.

The measures of the rhombuses angles are

`2\cdot 30^(\circ)=60^(\circ)\\ \\2\cdot 60^(\circ)=120^(\circ)`

Thus, option 4 is false.

The length of the longer diagonal is

`2x=2\cdot 2√(3)\approx 6.9\approx 7\ in`

Option 5 is true.