please explain how to do this question ---**Question 14**
**Question Stem:**
The area of trapezoid ABCD is 37.8, BE is 7cm, EC is 4 cm, Find the area of ABED.
**Figure Description:**
* **Type:** Geometric figure (Trapezoid with internal line segment).
* **Main Elements:**
* **Shape:** A quadrilateral labeled ABCD, appearing to be a trapezoid with parallel sides AD and BC.
* **Points:** Vertices A, B, C, D. Point E is located on the line segment BC, between B and C.
* **Lines:** Line segments AB, BC, CD, DA, and DE are drawn.
* **Labels:** Points A, B, C, D, E are labeled. The length of the segment BE is indicated as 7 (with a horizontal bracket below). The length of the segment EC is indicated as 4 (with a horizontal bracket below).
* **Relative Position:** A is above and to the left of B. D is above and to the right of C. E is on BC.
**Other Relevant Text:**
None for this specific question besides the problem statement and figure.
**Mathematical Formulas/Equations:**
None explicitly written, but the problem involves calculating area, likely using geometric formulas.
**Table Content:**
None.
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We have a trapezoid ABCD with total area thirty seven point eight square units. Point E lies on side BC, where BE equals seven centimeters and EC equals four centimeters. We need to find the area of quadrilateral ABED.
The key insight is to split the trapezoid into two parts: quadrilateral ABED shown in blue, and triangle DEC shown in red. Since both shapes share the same height from line AD to line BC, their area ratio equals their base ratio. Therefore, the ratio of area ABED to area DEC equals BE to EC, which is seven to four.
Now let's set up our equations. Let S one equal the area of ABED, and S two equal the area of triangle DEC. We know that S one plus S two equals thirty seven point eight. From our ratio, we have S one over S two equals seven over four. Therefore, S one equals seven fourths times S two.
Now let's solve the equations step by step. First, we substitute S one equals seven fourths times S two into our first equation. This gives us seven fourths times S two plus S two equals thirty seven point eight. Factoring out S two, we get S two times eleven fourths equals thirty seven point eight. Solving for S two, we get S two equals thirteen point seven five. Finally, S one equals seven fourths times thirteen point seven five, which equals twenty four point zero six. Therefore, the area of ABED is twenty four point zero six square units.
To summarize what we learned: We used the ratio method to split the trapezoid into two regions. The key insight is that area ratio equals base ratio when shapes share the same height. We set up a system of equations using the given total area and solved algebraically to find that the area of ABED equals twenty four point zero six square units.