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.

1) a = 60°

3) The perimeter of the rhombus is 16 inches.

5) The length of the longer diagonal is approximately 7 inches.

Step-by-step explanation:

From the attached diagram, we begin to compare the options

1)a = 60°

From the question, we are told that:

One interior angle is 30 degrees and another is a degrees

The sum of angles in a rhombus = 360°

There are 4 angles in a rhombus and each angle should normally be equal to 90° hence,

a = 90° - 30°

a = 60°

2) x = 3 in.

Looking at the attached diagram,

We solve for x using Pythagoras Theorem

a² + b² = c²

a = 2 inches

b = x = ??

c = 4 inches

Hence,

2² + b² = 4²

b² = 4² - 2²

b² = 16 - 4

b² = 12

b = √12

b = 3.4641016151 inches

Option 2 is incorrect

3) The perimeter of the rhombus is 16 inches.

A rhombus is a quadrilateral with 4 sides that are equal to each other.

The perimeter of a Rhombus = 4a

Where a = Length of the side of a rhombus

From the above question, we are told that, the length of each side = 4 inches

Hence, perimeter = 4 × 4

= 16 inches.

Option 3 is correct

4)The measure of the greater interior angle of the rhombus is 90°.

The measure of the greater interior angle = 2 × a°

a° has already been solved for in option 1 as 60°

Hence,

Measure of the greater interior angle = 2 × 60°

= 120°

Option 4 is not correct.

5)The length of the longer diagonal is approximately 7 inches.

Since the Rhombus forms 4 equal triangles, it means the diagonals are the same.

Hence, the length of the longer diagonal

= 2 × 3.4641016151

= 6.9282032303 inches

Approximately = 7 inches.

Option 5 is correct.

Hence, the correct options are

1) a = 60°

3) The perimeter of the rhombus is 16 inches. The measure of the greater interior angle of the rhombus is 90°.

5) The length of the longer diagonal is approximately 7 inches.

Explanation:

The set of whole numbers is {0, 1, 2, 3, ...}

So we include 0 and any positive number that doesn't have decimal or fractional components to it. Negative values are not included.

Whatever you pick for x, the y value must add to it to get 3. Both x and y must be numbers from the set of whole numbers.

There are four cases shown in the bullet points above, so we have four different solutions. We can't move onto x = 4 because the y solution would be y = -1, but -1 is not in the set of whole numbers.