In a trapezium ABCD, AB || DC and DC = 3AB. EF drawn parallel to AB which cuts AD at F AF 3 FD5. Diagonal DB = and BC at E such that intersects EF at G. Find the ratio of EF to AB.

Given:
- \(AB \parallel DC\) and \(DC = 3AB\)
- \(EF \parallel AB\) cuts \(AD\) at \(F\) with \(AF : FD = 3 : 5\)
- Diagonal \(DB\) intersects \(BC\) at \(E\) and \(EF\) at \(G\)

To find:
- The ratio of \(EF\) to \(AB\)

Let's assume:
- \(AB = x\)
- \(DC = 3x\)
- \(AF = 3k\) and \(FD = 5k\), so \(AD = AF + FD = 3k + 5k = 8k\)

Since \(EF \parallel AB\), \(EF\) is also a part of similar triangles formed by the intersection of diagonals.

Using the section formula for similar triangles:
- \(F\) divides \(AD\) in the ratio \(3:5\)
- Since \(EF \parallel AB\), \(EF\) will also divide \(DC\) in the same ratio \(3:5\)

Given the ratio \(3:5\), and knowing that \(EF\) divides \(DC\) in the same proportion as it divides \(AD\), we can use this proportionality to find the length of \(EF\).

Considering the lengths of segments:
- \(AB = x\)
- \(DC = 3x\)

Since \(DC\) is divided in the same ratio by \(EF\) as \(AD\) is by \(F\),
- \(DC = 3x\) will be divided into \(EF = 3/8\) of \(3x\) since \(EF\) corresponds to \(3\) parts of \(8\) parts.

Thus, the length of \(EF\) in terms of \(AB\):
\[ EF = \left(\frac{3}{8}\right) \times 3x = \frac{9x}{8} \]

Now, we find the ratio of \(EF\) to \(AB\):
\[ \text{Ratio of } EF \text{ to } AB = \frac{EF}{AB} = \frac{\frac{9x}{8}}{x} = \frac{9}{8} \]

Therefore, the ratio of \(EF\) to \(AB\) is \( \boxed{\frac{9}{8}} \).

Do you want further details or have any questions about this solution? 

Here are some additional questions you might find interesting:
1. How do you find the area of a trapezium given the lengths of the parallel sides and the height?
2. What is the relationship between similar triangles in a trapezium?
3. How can you use the properties of parallel lines to solve for unknown segments in trapezium?
4. How would you prove the triangles formed by intersecting diagonals are similar?
5. Can you use coordinate geometry to verify the ratio of the segments in the trapezium?

**Tip:** Always verify your final answer by cross-checking the properties and ratios used in the problem to ensure accuracy.

Explore how to find the ratio of segments EF to AB in a trapezium ABCD where AB is parallel to DC and DC is three times the length of AB. Learn about the use of similar triangles and the section formula theorem in geometry. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation