Certainly! Let's analyze and solve the problem step-by-step.
### Step 1: Understand the Problem
You have a quilt piece designed in the shape of a rhombus. This rhombus is made from four congruent triangles. One of the diagonals of the rhombus is equal to the side length of the rhombus.
### Step 2: Identify Given Information
- Let the side length of the rhombus be [tex]\( s \)[/tex].
- One of the diagonals is equal to the side length, which means diagonal 1 [tex]\( d_1 \)[/tex] = [tex]\( s \)[/tex].
### Step 3: Visualizing the Rhombus and its Properties
- Since all sides of a rhombus are equal, each side is [tex]\( s \)[/tex].
- A rhombus can be divided into four congruent right triangles by its diagonals.
- The diagonals of a rhombus bisect each other at right angles (90 degrees).
### Step 4: Analyzing the Right Triangles Formed
- When you draw the diagonals, you split the rhombus into four right triangles.
- Let the lengths of the diagonals be [tex]\( d_1 \)[/tex] and [tex]\( d_2 \)[/tex].
- Since [tex]\( d_1 = s \)[/tex], this half-diagonal is [tex]\( \frac{s}{2} \)[/tex].
### Step 5: Using the Pythagorean Theorem
Since we have right triangles:
- Hypotenuse (side of the rhombus) [tex]\( s \)[/tex].
- One leg [tex]\( \frac{s}{2} \)[/tex] (half of the first diagonal [tex]\( s \)[/tex]).
- Let [tex]\( \frac{d_2}{2} \)[/tex] be the other leg (half of the second diagonal that we need to find).
Using the Pythagorean theorem in one of the right triangles:
[tex]\[ s^2 = \left(\frac{s}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \][/tex]
### Step 6: Simplify and Solve for the Missing Diagonal
Substitute and solve for [tex]\( d_2 \)[/tex]:
[tex]\[ s^2 = \frac{s^2}{4} + \frac{d_2^2}{4} \][/tex]
Multiply everything by 4 to clear the fraction:
[tex]\[ 4s^2 = s^2 + d_2^2 \][/tex]
[tex]\[ 4s^2 - s^2 = d_2^2 \][/tex]
[tex]\[ 3s^2 = d_2^2 \][/tex]
[tex]\[ d_2 = \sqrt{3s^2} \][/tex]
[tex]\[ d_2 = s\sqrt{3} \][/tex]
### Step 7: Summarize Findings
- Side length, [tex]\( s \)[/tex]: [tex]\( 1 \)[/tex] (assuming [tex]\( s = 1 \)[/tex])
- Diagonal 1, [tex]\( d_1 \)[/tex]: [tex]\( 2 \)[/tex] (since it's twice [tex]\( \frac{s}{2} \)[/tex])
- Diagonal 2, [tex]\( d_2 \)[/tex]: [tex]\( 1.732 \)[/tex] (approximate value of [tex]\( s\sqrt{3} \)[/tex])
### Final Result
- Side length of the rhombus [tex]\( s \)[/tex]: [tex]\( 1 \)[/tex]
- Full length of one diagonal [tex]\( d_1 \)[/tex]: [tex]\( 2 \)[/tex]
- Full length of the other diagonal [tex]\( d_2 \)[/tex]: [tex]\( 1.732 \)[/tex]
These were the required values for the rhombus and its diagonals.
