Finding the Ratio of Top and Bottom Bases of a Trapezoid Given a Midpoint and Area Ratio

In a trapezoid ABCD, point E is the midpoint of AD and line segment EC divides the trapezoid into areas M and N with a ratio of 10:7. We aim to find the ratio of the top base AB (denoted by a) to the bottom base CD (denoted by b).

Step-by-Step Solution

Let's denote:

The top base of the trapezoid AB a The bottom base of the trapezoid CD b The height of the trapezoid h

The area of trapezoid ABCD is given by the formula:

Area ? × (a b) × h

Given that line segment EC divides the trapezoid into areas M and N with the ratio M:N 10:7, we can express the areas as:

M 10/17 × Area N 7/17 × Area

Now, let's delve into the solution with detailed calculations:

Breakdown of the Solution

The areas of triangles EBC and EAD can be determined as follows:

For triangle EBC:

Area of EBC ? × a × (h_1)

where h_1 is the height from E to line BC.

For triangle EAD:

Area of EAD ? × b × (h_2)

where h_2 is the height from E to line AD.

Since E is the midpoint, the heights h_1 and h_2 are proportional to the bases:

h_1 h × (b / (a b))

h_2 h × (a / (a b))

Using these heights, we can express the areas of M and N:

M ? × a × (h_1) ? × a × (h) × (b / (a b)) (ab / (2a b)) × h

N ? × b × (h_2) ? × b × (h) × (a / (a b)) (ab / (2a b)) × h

Applying the Area Ratio

Given the ratio M:N 10:7, we can set up the following equation:

(M/N) (10/7)

Substituting the area expressions:

((ab / (2a b)) × h) / ((ab / (2a b)) × h) 10/7

Simplifying:

(a × (b / (a b))) / (b × (a / (a b))) 10/7

This simplifies to:

a / b 10 / 7

Therefore, the ratio of the top base to the bottom base in trapezoid ABCD is:

Ratio of top base to bottom base 10 : 7

Conclusion

The problem is solved by leveraging the properties of the trapezoid and the given area ratio. This example demonstrates the application of geometric principles in solving complex figures and their divisions.

Keywords: Trapezoid, Midpoint, Area Ratio, Geometry