Same area, different shape
A median halves the area
Cut by a median
In triangle ABC, D is the midpoint of BC. The median AD cuts the triangle into two smaller triangles, ABD and ACD.
How do the areas of ABD and ACD compare?
What this lesson covers
The idea
A median divides a triangle into two triangles of equal area, since they have equal bases and the same height, even though they are in general not congruent.
Cut by a median
In triangle ABC, D is the midpoint of BC. The median AD cuts the triangle into two smaller triangles, ABD and ACD.
How do the areas of ABD and ACD compare?
- They are equal
- ABD is larger
- It depends on how the triangle looks
Move the vertex
Move A along the dotted line to places that are far apart. Watch the shapes of the two triangles and their areas.
A surprise
A median divides a triangle into two triangles of equal area, because they have equal bases and the same height, even though they are in general not congruent.
ABD and ACD have equal bases, BD = DC, and the same height h from A. Each has area 1/2 × BD × h, so the areas are equal. This is a surprise: the two triangles usually have different shapes, yet the same area.
Notes
A median divides a triangle into two triangles of equal area. They have equal bases and the same height, but in general they are not congruent.
Check yourself
A median divides triangle ABC, whose area is 48 cm², into ABD and ACD. What is the area of ABD?
Answer: 24 cm²
Each triangle has half of 48: 24 cm².
D is the midpoint of BC. Triangle ABD has base BD = 5 cm and height 6 cm. What is the area of the whole triangle ABC?
Answer: 30 cm²
ABD = 1/2 × 5 × 6 = 15. ACD is also 15. So ABC = 30 cm².
Why do ABD and ACD have the same area?
ABD and ACD have equal areas. Are they always congruent (exact copies)?
- Their three sides are equal in pairs. They usually have different sides. The areas are equal for another reason.
- They have equal bases (BD = DC) and the same height from A — correct. Yes. Area = 1/2 × base × height is the same for both.
- Both triangles are right-angled. They need not be right-angled. Equal bases and the same height are enough.
- Yes, equal area always means congruent. A long thin triangle and a short wide one can have the same area.
- No, they can never be congruent. They are congruent in special cases, such as when A is straight above D.
- No, they usually have different shapes — correct. Yes. They are exact copies only in special cases, for example when AB = AC.