A quilt piece is designed with four congruent triangles to …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

A quilt piece is designed with four congruent triangles to form a rhombus, where one diagonal equals the side length of the rhombus. Given that all sides are 4 inches long, the distance from the top point to the middle point is 2 inches, and 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 correct for the quilt piece? Select three options: a = 60°, x = 3 in, the perimeter of the rhombus is 16 inches, the greater interior angle of the rhombus is 90°, and the length of the longer diagonal is approximately 7 inches.

Short Answer

The process involves three main steps: first, finding angle ‘a’ as 60¬¨‚àû based on the property that angles in a rhombus sum to 360¬¨‚àû; second, calculating the perimeter as 16 inches using the formula for a rhombus; and finally, determining the longer diagonal, which is approximately 7 inches using the Pythagorean theorem.

Step-by-Step Solution

Step 1: Finding Angle ‘a’

The first step involves determining the value of angle a. For a rhombus, the property of angles states that the sum of all angles must equal 360°. Given one angle is 30°, we calculate a using the formula:

  • Angle sum in rhombus = 360¬¨‚àû
  • Remaining angles are: 90¬¨‚àû – 30¬¨‚àû
  • Thus, a = 60¬¨‚àû.

Step 2: Calculating the Perimeter

The second step is to calculate the perimeter of the rhombus. As a rhombus has all four sides equal, we can use the formula for the perimeter:

  • Perimeter = 4 ‚àöo length of one side
  • Since each side measures 4 inches, we compute:
  • Perimeter = 4 ‚àöo 4 = 16 inches.

Step 3: Determining the Longer Diagonal

The final step focuses on finding the length of the longer diagonal. In a rhombus, the diagonals intersect at right angles and bisect each other:

  • Based on the properties of rhombus, the length of the diagonal can be derived from the triangles formed by the diagonals.
  • Using Pythagorean theorem, you can approximate that the length of the longer diagonal is approximately 7 inches.

Related Concepts

Angle Sum

The total of all interior angles in a geometric shape, which for a rhombus equals 360°.

Perimeter

The total length of the sides of a shape, calculated for a rhombus as 4 times the length of one side.

Diagonal

A line segment that connects non-adjacent vertices of a polygon, with diagonals in a rhombus bisecting at right angles.

Table Of Contents
  1. A quilt piece is designed with four congruent triangles to form a rhombus, where one diagonal equals the side length of the rhombus. Given that all sides are 4 inches long, the distance from the top point to the middle point is 2 inches, and 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 correct for the quilt piece? Select three options: a = 60°, x = 3 in, the perimeter of the rhombus is 16 inches, the greater interior angle of the rhombus is 90°, and the length of the longer diagonal is approximately 7 inches.

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