Depth · one concept from first principles · step 2 of 4
One ■-th of the whole is the whole divided by ■
In the figure at the right, triangle DEC has the area of trapezoid ABCD. How long is segment EC, in ?
GeometryNumber system
Your answer
How to solve
Strategy Draw a Diagram — Go forwards to the trapezoid's area, take a fifth of it, then run the triangle formula backwards. The height is free -- the right-angle mark says the triangle stands on the same height as the trapezoid.
1STEP 1
Find the trapezoid's area
Add the two parallel sides, multiply by the height, and halve.
(11.8 + 18.6) × 9 ÷ 2 = 136.8
The trapezoid covers 136.8 cm².
6.G.A.1Draw A Diagram2STEP 2
Take a fifth of it
One fifth of an amount is that amount divided by 5.
136.8 ÷ 5 = 27.36
The triangle covers 27.36 cm².
6.NS.B.3Identify Subproblems3STEP 3
Run the triangle formula backwards
Its area is base times height divided by 2, and the height is the trapezoid's 9 cm, so double the area and divide by 9.
27.36 × 2 ÷ 9 = 6.08
EC is 6.08 cm.
6.G.A.1Work BackwardsTriangle DEC stands on the same height as the trapezoid.
Why?
D is on the trapezoid's top side and E is on its bottom side, and the right-angle mark says DE meets the bottom side square on, so DE is the distance between the two parallel sides.
🧱A transversal of parallel lines makes equal anglesA line crossing two parallel lines makes matching corresponding and alternate angles equal.
Why?
That distance is the same wherever it is measured, so it is the trapezoid's height and the triangle's height at once.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Answer
6.08 cm
Put it back: 6.08 × 9 ÷ 2 = 27.36, which is a fifth of 136.8.
Takeaway
A fraction of an area is just a division -- and a shape sitting on the same line shares the same height.
- Find the trapezoid's area
- Take a fifth of it
- Run the triangle formula backwards
Where next?
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