Problem
Equal heights make the area ratio the base ratio
Two triangles inside a larger one have equal areas. We need the ratio of one of them to a third triangle.
GeometryRatios
Your answer
How to solve
Strategy Convert to Algebra — The right angle hands over both heights at once: AE is 10 and DE is 4, both perpendicular to the base. Then the equal-area condition is one equation about how the base splits, and the answer follows from the two pieces.
1STEP 1
Find the two heights
The whole segment AE is AD plus DE, and the right angle at E makes both AE and DE perpendicular to the base.
6 + 4 = 10
Heights of 10 and 4 centimeters.
6.G.A.1Draw A Diagram2STEP 2
Write the equal areas as an equation
Call BE ■; then EC is 14 - ■. Triangle ABE is ■ times 10 halved, and triangle DEC is (14 - ■) times 4 halved.
5 × ■ = 2 × (14 - ■)
Halving both sides clears the 2s.
7.RP.A.2Convert To Algebra3STEP 3
Solve for the two parts of the base
Expanding gives 5 of the unknown on the left and 28 minus 2 of it on the right, so 7 of it is 28.
28 ÷ 7 = 4, 14 - 4 = 10
BE is 4 and EC is 10.
7.RP.A.2Work Backwards4STEP 4
Compare the two triangles asked about
Both stand on the base with the same height 10, so their areas are in the same ratio as their bases.
4 : 10 = 2 : 5
The ratio is 2 : 5.
7.RP.A.2Draw A DiagramTwo triangles on the same line with the same apex have areas in the ratio of their bases.
Why?
Both triangles stand on the same straight line and share the apex A, so the distance from A down to that line is the same height for both.
🧱A transversal of parallel lines makes equal anglesA line crossing two parallel lines makes matching corresponding and alternate angles equal.
Why?
A triangle's area is half its base times that height, so with the height the same the areas are just the bases scaled by one fixed number.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
2 : 5
Work the areas out: triangle ABE is 4 × 10 ÷ 2 = 20 and triangle DEC is 10 × 4 ÷ 2 = 20, equal as required; triangle AEC is 10 × 10 ÷ 2 = 50, and 20 : 50 is 2 : 5.
Takeaway
Triangles that share a height have areas in the same ratio as their bases.
- Find the two heights
- Write the equal areas as an equation
- Solve for the two parts of the base
- Compare the two triangles asked about