If `triangleABC ~ triangleDEF, BC= 3EF and ar(DABC) =117cm^2` find area `(triangleDEF). `

Answer

To find the area of triangle DEF given that triangle ABC is similar to triangle DEF, we can follow these steps:

### Step 1: Understand the relationship between the triangles
Since triangle ABC is similar to triangle DEF, the ratios of their corresponding sides are equal. This means:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \]

### Step 2: Use the given information
We are given that \( BC = 3 \times EF \). Therefore, we can express the ratio of the sides as:
\[ \frac{BC}{EF} = \frac{3}{1} \]

### Step 3: Relate the areas of the triangles
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore:
\[ \frac{Area(ABC)}{Area(DEF)} = \left(\frac{BC}{EF}\right)^2 \]
Substituting the known ratio:
\[ \frac{Area(ABC)}{Area(DEF)} = \left(\frac{3}{1}\right)^2 = 9 \]

### Step 4: Set up the equation
Let the area of triangle DEF be \( Area(DEF) \). We know that the area of triangle ABC is given as \( 117 \, cm^2 \). Thus, we can write:
\[ \frac{117}{Area(DEF)} = 9 \]

### Step 5: Solve for the area of triangle DEF
Cross-multiplying gives:
\[ 117 = 9 \times Area(DEF) \]
Now, divide both sides by 9:
\[ Area(DEF) = \frac{117}{9} = 13 \, cm^2 \]

### Final Answer
The area of triangle DEF is \( 13 \, cm^2 \).

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