Try each problem before reading the solutions. Write down the step that makes your argument complete.
Problems
1. A median joins vertex A of triangle ABC to the midpoint of BC. Prove that it divides the area in half.
2. Points D and E on BC divide it into three equal segments in the order B, D, E, C. If triangle ABC has area 45, find the area of triangle ADE.
3. D lies on AB and E on AC, with AD:DB = 1:2 and AE:EC = 2:1. If triangle ABC has area 54, find the area of triangle ADE and justify your calculation.
Solutions
1. Both smaller triangles have the same altitude from A and equal bases on BC, so their areas are equal.
2. The triangles ADE and ABC have the same altitude from A, and DE is one third of BC. The area is 45 ÷ 3 = 15.
3. The area is 12. Triangles ADE and ABE share the altitude from E, so their area ratio is AD/AB = 1/3. Triangles ABE and ABC share the altitude from B when their bases lie on AC, so their ratio is AE/AC = 2/3. Multiply 54 by 1/3 and 2/3.
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