To find the length of the corresponding side of the second triangle, we need to use the relationship between the areas of similar triangles and the lengths of their corresponding sides.
Step 1: Understand the relationship between the areas and the sides of similar triangles.
For two similar triangles, the ratio of their areas is equal to the square of the ratio of the lengths of their corresponding sides. Mathematically, it can be expressed as:
[tex]\[ \text{Area}_1 / \text{Area}_2 = (\text{Side}_1 / \text{Side}_2)^2 \][/tex]
Step 2: Assign the given values to the variables.
Let's denote the areas of the triangles and the sides as follows:
- Area of the first triangle ([tex]\(\text{Area}_1\)[/tex]) = 9 cm²
- Area of the second triangle ([tex]\(\text{Area}_2\)[/tex]) = 49 cm²
- Length of a side of the first triangle ([tex]\(\text{Side}_1\)[/tex]) = 2 cm
- Length of the corresponding side of the second triangle ([tex]\(\text{Side}_2\)[/tex]) = ?
Step 3: Write the equation based on the relationship.
[tex]\[ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{\text{Side}_1}{\text{Side}_2}\right)^2 \][/tex]
Substituting the given values:
[tex]\[ \frac{9}{49} = \left(\frac{2}{\text{Side}_2}\right)^2 \][/tex]
Step 4: Solve for [tex]\(\text{Side}_2\)[/tex].
First, we'll rearrange the equation to isolate [tex]\(\text{Side}_2\)[/tex].
[tex]\[ \left(\frac{2}{\text{Side}_2}\right)^2 = \frac{9}{49} \][/tex]
Taking the square root of both sides to solve for [tex]\(\text{Side}_2\)[/tex]:
[tex]\[ \frac{2}{\text{Side}_2} = \sqrt{\frac{9}{49}} \][/tex]
[tex]\[ \frac{2}{\text{Side}_2} = \frac{3}{7} \][/tex]
Inverting both sides to solve for [tex]\(\text{Side}_2\)[/tex]:
[tex]\[ \text{Side}_2 = \frac{2 \cdot 7}{3} \][/tex]
[tex]\[ \text{Side}_2 = \frac{14}{3} \][/tex]
[tex]\[ \text{Side}_2 = 4.\overline{66} \][/tex] cm (approximately 4.67 cm)
Step 5: Conclude the answer.
So, the length of the corresponding side of the second triangle is approximately 4.67 cm.
