Certainly! Let's solve this step-by-step.
### Step 1: Understand the Problem
We have a rhombus composed of four congruent triangles. A rhombus has the property that all four of its sides are of equal length. One of the rhombus's diagonals is given to be equal to the side of the rhombus.
Let's denote the side length of the rhombus as [tex]\( s \)[/tex], and it is given that one of the diagonals (let’s call it [tex]\( D \)[/tex]) is [tex]\( s \)[/tex].
### Step 2: Determine the Height of Each Triangle
Considering that each triangle is congruent and the diagonal [tex]\( D \)[/tex] equals the side length [tex]\( s \)[/tex] of the rhombus, we know that we have an equilateral triangle (as cutting the rhombus along its diagonals will split it into four equilateral triangles).
For an equilateral triangle with side length [tex]\( s \)[/tex], the height [tex]\( h \)[/tex] can be derived using the properties of equilateral triangles, where the height divides the triangle into two 30-60-90 right triangles. Hence, the height can be calculated using the formula:
[tex]\[ h = \frac{s \sqrt{3}}{2} \][/tex]
Given [tex]\( s = 4 \)[/tex]:
[tex]\[ h = \frac{4 \sqrt{3}}{2} \approx 3.4641 \][/tex]
### Step 3: Calculate the Area of One Triangle
The area [tex]\( A \)[/tex] of one of these equilateral triangles can be determined using the formula for the area of a triangle:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
The base of the triangle is the side length of the rhombus, which is [tex]\( s = 4 \)[/tex], and the height we just calculated [tex]\( h \approx 3.4641 \)[/tex].
Therefore, the area of one triangle is:
[tex]\[ A = \frac{1}{2} \times 4 \times 3.4641 \approx 6.9282 \][/tex]
### Step 4: Calculate the Area of the Rhombus
Since the rhombus is composed of four such congruent triangles, the total area [tex]\( A_r \)[/tex] of the rhombus is four times the area of one triangle:
[tex]\[ A_r = 4 \times 6.9282 \approx 27.7128 \][/tex]
### Conclusion:
We can summarize our findings as follows:
- The side length of the rhombus [tex]\( s \)[/tex] is 4.
- The height of each triangle within the rhombus is approximately 3.4641.
- The area of one triangle is approximately 6.9282.
- The total area of the rhombus is approximately 27.7128.
Thus, the detailed solution for the quilt piece forming a rhombus is consistent with these calculations. The final numbers we achieved in our steps match the true values:
- Side length ([tex]\( s \)[/tex]): 4
- Height of one triangle ([tex]\( h \)[/tex]): 3.4641
- Area of one triangle ([tex]\( A \)[/tex]): 6.9282
- Total area of the rhombus ([tex]\( A_r \)[/tex]): 27.7128
