The area of two similar triangles PQR and XYZ are `144 cm^(2)` and `49 cm^(2)` respectively. If the shortest side of larger `DeltaPQR` be 24 cm, then find the shortest side of the smaller triangle XYZ.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the areas of similar triangles.
The areas of two similar triangles are proportional to the square of the lengths of their corresponding sides.

### Step 2: Set up the ratio of the areas.
Let the area of triangle PQR be \( A_1 = 144 \, \text{cm}^2 \) and the area of triangle XYZ be \( A_2 = 49 \, \text{cm}^2 \). The ratio of the areas can be expressed as:
\[ \frac{A_1}{A_2} = \frac{144}{49} \]

### Step 3: Relate the ratio of the areas to the ratio of the sides.
Since the triangles are similar, we can express the ratio of the sides as the square root of the ratio of the areas:
\[ \frac{PQ}{XY} = \sqrt{\frac{A_1}{A_2}} = \sqrt{\frac{144}{49}} = \frac{12}{7} \]

### Step 4: Substitute the known value of the shortest side.
We know that the shortest side of triangle PQR (denoted as \( PQ \)) is \( 24 \, \text{cm} \). We can substitute this value into the ratio:
\[ \frac{24}{XY} = \frac{12}{7} \]

### Step 5: Solve for the shortest side of triangle XYZ.
Cross-multiply to solve for \( XY \):
\[ 24 \cdot 7 = 12 \cdot XY \]
\[ 168 = 12 \cdot XY \]
Now, divide both sides by 12:
\[ XY = \frac{168}{12} = 14 \, \text{cm} \]

### Conclusion:
The shortest side of the smaller triangle XYZ is \( 14 \, \text{cm} \).

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