Question

In △ABC, point M is the midpoint of AB, point D is the midpoint of CM, and ABMD = 3 cm². Find ACDB, AAMC, AABD, AADC, and AABC.

a) 3 cm², 6 cm², 6 cm², 12 cm²
b) 3 cm², 6 cm², 12 cm², 12 cm²
c) 6 cm², 12 cm², 6 cm², 12 cm²
d) 3 cm², 12 cm², 6 cm², 12 cm²

Answer 1

In triangle ABC with various midpoints, the areas of the sub-triangles can be found using the formula for the area of a triangle. The areas are: AAMB = 3/4 cm², AABD = 3/2 cm², AADC = 3 cm², AABC = 3 cm².

Step-by-step explanation:

In triangle ABC, point M is the midpoint of AB, point D is the midpoint of CM, and ABMD = 3 cm². To find the areas of the sub-triangles, we can use the fact that the area of a triangle is one-half the product of its base and height.

AAMB: Since M is the midpoint of AB, the base of AAMB is AB and the height is MD. So, AAMB = (1/2) * AB * MD = (1/2) * AB * (1/2) * CM = (1/4) * AB * CM = (1/4) * 3 cm² = 3/4 cm².

AABD: The base of AABD is AB and the height is the same as ABMD. So, AABD = (1/2) * AB * ABMD = (1/2) * 3 cm² = 3/2 cm².

AADC: The base of AADC is AC and the height is the same as CM. So, AADC = (1/2) * AC * CM = (1/2) * 2 * 3 = 3 cm².

AABC: The base of AABC is AB and the height is the same as AC. So, AABC = (1/2) * AB * AC = (1/2) * 3 * 2 = 3 cm².

Therefore, the areas of the sub-triangles are: AAMB = 3/4 cm², AABD = 3/2 cm², AADC = 3 cm², AABC = 3 cm².