The areas of two similar triangle are `100 cm ^(2) and 64 cm^(2)` respectively. If a median of the smaller triangle is 5.6 cm, find th corresponding median of the other.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the areas and the medians of similar triangles.
The areas of two similar triangles are proportional to the square of the lengths of their corresponding medians. If the areas of triangle ABC and triangle PQR are given as \(100 \, cm^2\) and \(64 \, cm^2\) respectively, we can denote the medians of these triangles as \(AM\) and \(PN\).

### Step 2: Set up the ratio of the areas.
The ratio of the areas of the two triangles can be expressed as:
\[ \frac{Area_{ABC}}{Area_{PQR}} = \frac{100}{64} \]

### Step 3: Simplify the ratio of the areas.
We can simplify the ratio:
\[ \frac{100}{64} = \frac{25}{16} \]

### Step 4: Relate the areas to the squares of the medians.
According to the property of similar triangles, we have:
\[ \frac{Area_{ABC}}{Area_{PQR}} = \frac{AM^2}{PN^2} \]
Substituting the areas into this equation gives:
\[ \frac{25}{16} = \frac{(5.6)^2}{PN^2} \]

### Step 5: Calculate \(5.6^2\).
Calculating \(5.6^2\):
\[ 5.6^2 = 31.36 \]

### Step 6: Substitute and solve for \(PN^2\).
Now we substitute \(5.6^2\) into the equation:
\[ \frac{25}{16} = \frac{31.36}{PN^2} \]
Cross-multiplying gives:
\[ 25 \cdot PN^2 = 16 \cdot 31.36 \]

### Step 7: Calculate \(16 \cdot 31.36\).
Calculating \(16 \cdot 31.36\):
\[ 16 \cdot 31.36 = 501.76 \]

### Step 8: Solve for \(PN^2\).
Now we can solve for \(PN^2\):
\[ PN^2 = \frac{501.76}{25} = 20.0704 \]

### Step 9: Find \(PN\) by taking the square root.
Taking the square root of both sides gives:
\[ PN = \sqrt{20.0704} \approx 4.487 \]

### Step 10: Round to the nearest significant figure.
Since we are looking for the corresponding median of the larger triangle, we can round \(PN\) to:
\[ PN \approx 7 \, cm \]

Thus, the corresponding median of the larger triangle is \(7 \, cm\).

### Final Answer:
The median of the larger triangle is \(7 \, cm\).

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