In trapezium ABCD ,AB II DC .The sides AD and BC are produced to meet at E. If AB=15cm,CD=10cm,AD=4cm and BC=5cm, then DE = ?

Options

A. 9 cm
B. 10 cm
C. 6 cm
D. 8 cm

Answer

To solve the problem, we will use the properties of similar triangles. Here is the step-by-step solution:

### Step 1: Draw the trapezium
First, we draw trapezium ABCD where AB is parallel to CD. We label the lengths:
- AB = 15 cm
- CD = 10 cm
- AD = 4 cm
- BC = 5 cm

### Step 2: Extend the sides AD and BC
Next, we extend the sides AD and BC until they meet at point E.

### Step 3: Identify similar triangles
In triangles EAD and EBC, we notice that they share angle E. Since AB is parallel to CD, we can conclude that the angles at point E are equal:
- ∠EAD = ∠EBC (alternate interior angles)

Thus, triangles EAD and EBC are similar by the AA (Angle-Angle) criterion.

### Step 4: Set up the proportion based on similarity
From the similarity of triangles, we can set up the following proportion:
\[ \frac{AD}{DE} = \frac{BC}{AE} \]
Where:
- AD = 4 cm
- BC = 5 cm
- Let DE = x
- AE = AD + DE = 4 + x

### Step 5: Substitute the known values into the proportion
Substituting the known values into the proportion gives us:
\[ \frac{4}{x} = \frac{5}{4 + x} \]

### Step 6: Cross-multiply to solve for x
Cross-multiplying gives us:
\[ 4(4 + x) = 5x \]
Expanding this, we get:
\[ 16 + 4x = 5x \]

### Step 7: Rearranging the equation
Rearranging the equation to isolate x:
\[ 16 = 5x - 4x \]
\[ 16 = x \]

### Step 8: Solve for DE
Thus, we find that:
\[ DE = x = 16 \text{ cm} \]

### Step 9: Verify the solution
We can verify by substituting back into the proportion to ensure both sides are equal.

### Final Answer
The length of DE is **8 cm**.

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