Figure ABCD is a trapezoid with DC = (1/3) AB. What is the ratio of area ABCD to area DCE? Provide an argument to support your answer.
Could it be
1:3 ?
Can you please explain?
To determine the ratio of the areas of trapezoid ABCD to triangle DCE, we need to find the areas of both shapes first. Let's break it down step-by-step:
1. Given that DC is equal to one-third (1/3) of AB, we can write DC = (1/3)AB.
2. Notice that triangle DCE is formed by drawing a line segment from point C to point E, where E is the intersection of AB and DC.
3. Since DC is (1/3) of AB, we can also say that AB is 3 times DC, or AB = 3DC.
4. Now, we can assign a variable to the length of DC. Let's call it x. Therefore, AB would be 3x.
5. The area of trapezoid ABCD is given by the formula:
Area_ABCD = [(sum of the lengths of the bases) / 2] * height.
In this case, the bases of the trapezoid are AB and DC, and the height is the perpendicular distance between AB and DC.
The sum of the bases is AB + DC, which can be written as 3x + x = 4x. Since AB and DC constitute the entire top and bottom lengths of the trapezoid, the sum of the bases is equivalent to the length of the top side of the trapezoid (as they are parallel).
Thus, the area of trapezoid ABCD is: Area_ABCD = [(4x) / 2] * height.
6. The area of triangle DCE is given by the formula:
Area_DCE = (base * height) / 2.
In this case, the base of the triangle is DC (which is x) and the height is the perpendicular distance between DC and AB.
Thus, the area of triangle DCE is: Area_DCE = (x * height) / 2.
7. To determine the ratio of the areas, we divide the area of trapezoid ABCD by the area of triangle DCE:
Ratio = Area_ABCD / Area_DCE
Substituting the area formulas we derived earlier, we get:
Ratio = [(4x) / 2] * height / [(x * height) / 2]
= 2 * 4x / 2 * x
= 8x / 2x
= 8/2
= 4/1
= 4:1
Therefore, the correct ratio is 4:1, not 1:3 as you suggested.