If `triangleABC ~ triangleDEF, ar(triangleDEF) =100cm^(2) and (AB)/(DE)=1/2," then ar(DABC) is"`

Options

A. `50cm^(2)`
B. `25cm^(2)`
C. `4cm^(2)`
D. `200cm^(2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between the two triangles
Given that triangle ABC is similar to triangle DEF, we know that the ratio of their areas is equal to the square of the ratio of their corresponding sides.

### Step 2: Write the formula for the ratio of areas
The formula for the ratio of the areas of two similar triangles is:
\[ \frac{ar(triangle ABC)}{ar(triangle DEF)} = \left(\frac{AB}{DE}\right)^2 \]

### Step 3: Substitute the known values
We know:
- \( ar(triangle DEF) = 100 \, cm^2 \)
- \( \frac{AB}{DE} = \frac{1}{2} \)

Substituting these values into the formula gives:
\[ \frac{ar(triangle ABC)}{100} = \left(\frac{1}{2}\right)^2 \]

### Step 4: Calculate the square of the ratio of sides
Calculating the square of \( \frac{1}{2} \):
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 5: Set up the equation
Now we can set up the equation:
\[ \frac{ar(triangle ABC)}{100} = \frac{1}{4} \]

### Step 6: Solve for \( ar(triangle ABC) \)
To find \( ar(triangle ABC) \), multiply both sides by 100:
\[ ar(triangle ABC) = 100 \times \frac{1}{4} = 25 \, cm^2 \]

### Final Answer
Thus, the area of triangle ABC is:
\[ ar(triangle ABC) = 25 \, cm^2 \]

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