If DE Is parallel to BC and bisects the other two sides of the triangle ABC such that the ratio `(AD)/(DB) = 5/13` and the length of the part EC is 26 cm. Then deter mine the length of AE (in cm).

Options

A. 5
B. 10
C. 15
D. 20

Answer

To solve the problem, we will use the properties of similar triangles and the given ratios. Here’s a step-by-step solution:

### Step 1: Understand the triangle and the parallel line
We have triangle ABC with DE parallel to BC. According to the properties of similar triangles, the ratios of the lengths of corresponding sides are equal.

### Step 2: Set up the ratios
Given that \( \frac{AD}{DB} = \frac{5}{13} \), we can denote:
- \( AD = 5x \)
- \( DB = 13x \)

### Step 3: Find the total length of AB
The total length of side AB can be calculated as:
\[ AB = AD + DB = 5x + 13x = 18x \]

### Step 4: Use the given information about EC
We are given that \( EC = 26 \) cm. Since DE is parallel to BC, we can set up the ratio of the segments:
\[ \frac{AD}{DB} = \frac{AE}{EC} \]

### Step 5: Substitute the known values into the ratio
We know:
\[ \frac{AD}{DB} = \frac{5}{13} \quad \text{and} \quad EC = 26 \]
Thus, we can write:
\[ \frac{5}{13} = \frac{AE}{26} \]

### Step 6: Cross-multiply to solve for AE
Cross-multiplying gives us:
\[ 5 \cdot 26 = 13 \cdot AE \]
\[ 130 = 13 \cdot AE \]

### Step 7: Solve for AE
Now, divide both sides by 13:
\[ AE = \frac{130}{13} = 10 \text{ cm} \]

### Conclusion
The length of AE is 10 cm.

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