Let's analyze the problem step-by-step.
1. Properties of a Rhombus:
- All sides of a rhombus are of equal length.
- The diagonals bisect each other at right angles.
- Each diagonal splits the rhombus into two congruent triangles.
2. Given Information:
- One of the diagonals (let’s refer to it as d1) is equal to the side length of the rhombus (s).
- Assume the side length of the rhombus (s) is 10 units. This is a specific numerical value, and you can use other values based on the problem context.
3. Goal:
- Calculate the length of the other diagonal (d2) and the area of the rhombus.
4. Visualize and Setup the Right Triangle:
- Since d1 is equal to the side length of the rhombus (s), we know that each half of this diagonal forms a right-angled triangle with half of the other diagonal (d2/2) and the given side (s).
- By drawing these diagonals, the halves create four right-angled triangles inside the rhombus.
5. Applying the Pythagorean Theorem:
- For one of the right triangles formed by half of d1 and half of d2, we have:
- (s/2)² + (d2/2)² = s²
6. Solve for d2:
- Substitute the specific side length (s = 10 units):
[tex]\[
\left(\frac{10}{2}\right)^2 + \left(\frac{d2}{2}\right)^2 = 10^2
\][/tex]
[tex]\[
5^2 + \left(\frac{d2}{2}\right)^2 = 100
\][/tex]
[tex]\[
25 + \left(\frac{d2}{2}\right)^2 = 100
\][/tex]
[tex]\[
\left(\frac{d2}{2}\right)^2 = 100 - 25
\][/tex]
[tex]\[
\left(\frac{d2}{2}\right)^2 = 75
\][/tex]
[tex]\[
\frac{d2}{2} = \sqrt{75}
\][/tex]
[tex]\[
\frac{d2}{2} = 5\sqrt{3}
\][/tex]
[tex]\[
d2 = 10\sqrt{3}
\][/tex]
- Numerically: [tex]\( d2 \approx 10 \times 1.732 = 17.32050807568877\)[/tex]
7. Calculate the Area of the Rhombus:
- The area of the rhombus can be found using the diagonals:
[tex]\[
\text{Area} = \frac{d1 \times d2}{2}
\][/tex]
- Substitute the values [tex]\( d1 = 10 \)[/tex] units and [tex]\( d2 \approx 17.32050807568877 \)[/tex] units:
[tex]\[
\text{Area} \approx \frac{10 \times 17.32050807568877}{2}
\][/tex]
[tex]\[
\text{Area} \approx \frac{173.2050807568877}{2}
\][/tex]
[tex]\[
\text{Area} \approx 86.60254037844385
\][/tex]
Summary:
- Side length of the rhombus (s): 10 units
- Length of diagonal d1: 10 units
- Length of diagonal d2: [tex]\(10\sqrt{3} \approx 17.32050807568877\)[/tex] units
- Area of the rhombus: [tex]\( \approx 86.60254037844385\)[/tex] square units
Therefore, by understanding the problem's structure and applying geometric properties and the Pythagorean theorem, these values can be derived for the given quilt piece resembling a rhombus.
