Areas of two similar triangles are \(36\,\text{cm}^2\) and \(100\,\text{cm}^2\). If a corresponding side of the larger is 20 cm, find the corresponding side of the smaller.
12 cm
Step 1: Write the areas of the triangles.
Smaller triangle area = \(36\,\text{cm}^2\)
Larger triangle area = \(100\,\text{cm}^2\)
Step 2: Recall the rule for similar triangles.
If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
That is: \( \dfrac{\text{Area of smaller}}{\text{Area of larger}} = \left(\dfrac{\text{Side of smaller}}{\text{Side of larger}}\right)^2 \)
Step 3: Substitute the known values.
\( \dfrac{36}{100} = \left(\dfrac{\text{Side of smaller}}{20}\right)^2 \)
Step 4: Simplify the fraction.
\( \dfrac{36}{100} = \dfrac{9}{25} \)
So, \( \left(\dfrac{\text{Side of smaller}}{20}\right)^2 = \dfrac{9}{25} \)
Step 5: Take square root on both sides.
\( \dfrac{\text{Side of smaller}}{20} = \dfrac{3}{5} \)
Step 6: Multiply both sides by 20 cm.
\( \text{Side of smaller} = 20 \times \dfrac{3}{5} = 12\,\text{cm} \)
Final Answer: The corresponding side of the smaller triangle is 12 cm.