Solving for the Area of Triangle DEX in a Given Geometric Configuration

In the geometric configuration of a triangle ABC, let D, E, and F be points on sides BC, AC, and AB, respectively, with given ratios BD:DC 1:1, CE:EA 1:3, and AF:FB 1:4. A line parallel to AB is drawn from D to G on side AC. These elements meet at point X where DG and EF intersect. We are tasked with determining the area of triangle DEX.

Step-by-Step Solution

To find the area of triangle DEX, we will follow a series of steps based on geometric properties and calculations.

Step 1: Determine the Area of Triangle ABC

Given that the area of triangle ABC is 120 units, we need to calculate the areas of other related triangles to solve for the area of triangle DEX.

Step 2: Identify Midpoints and Use the Midpoint Theorem

Point D is the midpoint of BC, and line DG is parallel to AB. According to the Midpoint Theorem, if a line is drawn parallel to one side of a triangle, it divides the other two sides into proportional segments. Therefore, G is also the midpoint of AC because DG is parallel to AB.

Area of Triangle GCD: - Since G is the midpoint of AC, the area of triangle GCD is half of the area of triangle ABC. Thus, the area of GCD is ( frac{1}{2} times 120 60 ) square units. - Triangle GCD is one-fourth of the area of triangle ABC because D is the midpoint of BC. This confirms the area of GCD to be ( frac{120}{4} 30 ) square units.

Step 3: Determine the Area of Triangle EDG

Since ED is the median of triangle GCD, it divides triangle GCD into two equal areas. Therefore, the area of triangle EDG is ( frac{1}{2} times 30 15 ) square units.

Area of Triangle EDG: - ED is a median of triangle GCD, so the area of triangle EDG is ( frac{1}{2} times 30 15 ).

Step 4: Calculate the Area of Triangle AEB

Consider triangle AEB. Since BE is a median of triangle GCB, and BC is divided into two equal parts by D, the area of triangle AEB is 75% of the area of triangle ABC. The area of triangle AEB is calculated as follows:

1. The area of triangle GCB is ( frac{120}{2} 60 ) square units since G is the midpoint of AC and C is the midpoint of BC.

2. The area of triangle ECB is half of triangle GCB, which is ( frac{60}{2} 30 ) square units.

3. The area of triangle AEB is the area of triangle ABC minus the area of triangle ECB, which is ( 120 - 30 90 ) square units.

Step 5: Find the Area of Triangle EAF

Triangle EAF is part of triangle AEB and its area is one-fifth of triangle AEB:

[ text{Area of } triangle EAF frac{1}{5} times 90 18 text{ square units} ]

Step 6: Determine the Area of Triangle EGX

Since triangle EAF is similar to triangle EGX and GE is one-third of AE, the area of triangle EGX is one-ninth of the area of triangle EAF:

[ text{Area of } triangle EGX frac{1}{9} times 18 2 text{ square units} ]

Step 7: Calculate the Area of Triangle DEX

The area of triangle DEX is the area of triangle EDG minus the area of triangle EGX:

[ text{Area of } triangle DEX 15 - 2 13 text{ square units} ]

Thus, the area of triangle DEX is ( boxed{13} ) square units.

Conclusion

The solution to the problem of finding the area of triangle DEX involves a series of logical and geometric deductions based on the given ratios and the properties of triangles. By carefully analyzing each step and using geometric principles, we were able to arrive at the correct area for triangle DEX.

Keywords: area of triangle, geometric configuration, parallel lines