Answer :
To solve the problem of finding the area of a rhombus composed of four congruent triangles, where one of the diagonals is equal to the side length of the rhombus, we can follow these steps.
### Step-by-Step Solution
1. Understand the Properties of the Rhombus and its Triangles:
- A rhombus is a quadrilateral with all sides of equal length.
- It can be divided into four congruent (identical in shape and size) triangles by its diagonals.
- One of the given diagonals is equal to the side length of the rhombus.
2. Notation and Side Length:
- Let [tex]\( s \)[/tex] represent the side length of the rhombus.
- Given that one of the diagonals is equal to the side length [tex]\( s \)[/tex].
3. Formulating the Problem:
- The area (A) of a rhombus can be calculated using one of its standard properties: [tex]\( A = \frac{1}{2} \times d_1 \times d_2 \)[/tex], where [tex]\( d_1 \)[/tex] and [tex]\( d_2 \)[/tex] are the lengths of the diagonals.
- Here, for simplicity, we assume [tex]\( s = 1 \)[/tex] (the unit doesn't matter as we will ultimately consider proportions).
4. Area of One Triangle:
- Each triangle in the rhombus is a right triangle if one diagonal equals a side length.
- The area of one triangle can be formulated as [tex]\( \text{Triangle Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
- Since one diagonal equals the side length, the base of one of the triangles is [tex]\( s \)[/tex] and the height is half of the other diagonal.
- Therefore, the area of one triangle is [tex]\( \text{Area of one Triangle} = \frac{1}{2} \times s \times \frac{s}{2} = \frac{1}{2} \times 1 \times \frac{1}{2} = 0.25 \)[/tex].
5. Total Area of the Rhombus:
- Since the rhombus consists of 4 such congruent triangles, the total area is given by:
- [tex]\( \text{Total Area} = 4 \times \text{Area of one Triangle} = 4 \times 0.25 = 1.0 \)[/tex].
The calculations lead us to believe that the area of the rhombus is:
- The area of one triangle: [tex]\( 0.25 \)[/tex]
- The total area of the four triangles (and hence the rhombus): [tex]\( 1.0 \)[/tex]
Thus, the final solution is:
[tex]\[ \text{Triangle area} = 0.25 \][/tex]
[tex]\[ \text{Total triangle area} = 1.0 \][/tex]
[tex]\[ \text{Rhombus area} = 1.0 \][/tex]
### Step-by-Step Solution
1. Understand the Properties of the Rhombus and its Triangles:
- A rhombus is a quadrilateral with all sides of equal length.
- It can be divided into four congruent (identical in shape and size) triangles by its diagonals.
- One of the given diagonals is equal to the side length of the rhombus.
2. Notation and Side Length:
- Let [tex]\( s \)[/tex] represent the side length of the rhombus.
- Given that one of the diagonals is equal to the side length [tex]\( s \)[/tex].
3. Formulating the Problem:
- The area (A) of a rhombus can be calculated using one of its standard properties: [tex]\( A = \frac{1}{2} \times d_1 \times d_2 \)[/tex], where [tex]\( d_1 \)[/tex] and [tex]\( d_2 \)[/tex] are the lengths of the diagonals.
- Here, for simplicity, we assume [tex]\( s = 1 \)[/tex] (the unit doesn't matter as we will ultimately consider proportions).
4. Area of One Triangle:
- Each triangle in the rhombus is a right triangle if one diagonal equals a side length.
- The area of one triangle can be formulated as [tex]\( \text{Triangle Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
- Since one diagonal equals the side length, the base of one of the triangles is [tex]\( s \)[/tex] and the height is half of the other diagonal.
- Therefore, the area of one triangle is [tex]\( \text{Area of one Triangle} = \frac{1}{2} \times s \times \frac{s}{2} = \frac{1}{2} \times 1 \times \frac{1}{2} = 0.25 \)[/tex].
5. Total Area of the Rhombus:
- Since the rhombus consists of 4 such congruent triangles, the total area is given by:
- [tex]\( \text{Total Area} = 4 \times \text{Area of one Triangle} = 4 \times 0.25 = 1.0 \)[/tex].
The calculations lead us to believe that the area of the rhombus is:
- The area of one triangle: [tex]\( 0.25 \)[/tex]
- The total area of the four triangles (and hence the rhombus): [tex]\( 1.0 \)[/tex]
Thus, the final solution is:
[tex]\[ \text{Triangle area} = 0.25 \][/tex]
[tex]\[ \text{Total triangle area} = 1.0 \][/tex]
[tex]\[ \text{Rhombus area} = 1.0 \][/tex]
