Ratios of Triangle Areas (When the Heights Are Equal)
The area of a triangle is (1/2) × base × height. If either the bases or the heights are equal, the ratio of the other quantities directly determines the ratio of the areas.
Basics of Area Ratios
- Two triangles with equal heights → the ratio of their areas remains the “ratio of their bases”
- Two triangles with equal bases → the ratio of their areas remains the “ratio of their heights”
- Always keep “area = (1/2) × base × height” in mind
Example 1: △ABD and △CBD have equal heights. If BD = 6 and DC = 4, find the area ratio △ABD : △CBD.
Solution: Since the heights are equal, area ratio = base ratio = BD : DC = 6 : 4 = 3 : 2
Solution: Since the heights are equal, area ratio = base ratio = BD : DC = 6 : 4 = 3 : 2
Example 2: Point M is taken on side BC of △ABC, dividing it so that BM : MC = 2 : 3. What is the area ratio of △ABM and △AMC?
Solution: The height from A to BC is the same for both triangles. Therefore, area ratio = base ratio = BM : MC = 2 : 3
Solution: The height from A to BC is the same for both triangles. Therefore, area ratio = base ratio = BM : MC = 2 : 3
A “median” (a line segment connecting a vertex to the midpoint of the opposite side) divides a triangle into two triangles with equal areas.