Final answer:
The area of triangles CDB, AMC, ABD, ADC, and ABC are 12cm², 12cm², 2√6cm², 2√6cm², and 4√6cm² respectively.
Step-by-step explanation:
Let's begin by finding the area of triangle BMD. We know that D is the midpoint of CM, so DM is equal to MC. Since M is the midpoint of AB, we also know that BM is equal to MA. Therefore, we can conclude that DM = MC = BM = MA.
Since triangle BMD is isosceles, the height of the triangle can be found by drawing a perpendicular line from D to BM. Let's call this perpendicular line DH. By the properties of isosceles triangles, DH is also the median and the altitude of triangle BMD. Since D is the midpoint of CM, DH is half of the height of CM, so the height of triangle BMD is DH = 0.5 * CM.
Now let's use the formula for the area of a triangle, which is given by A = 0.5 * base * height. The base of triangle BMD is BM, which is equal to MA, and the height is DH. Therefore, the area of triangle BMD is A = 0.5 * BM * DH = 0.5 * MA * 0.5 * CM = 0.25 * CM * MA.
We are given that the area of triangle BMD is 3 cm², so we can set up the equation 0.25 * CM * MA = 3. Now let's find the values of CM and MA. Since CM is the same as DM, which is half of the length of CM, we can let CM = 2x. Similarly, MA is the same as BM, which is half of the length of AB. Since M is the midpoint of AB, MA = BM = 0.5 * AB = 0.5 * 2x = x.
Plugging in these values into our equation, we get 0.25 * (2x) * x = 3. Simplifying, we have 0.5x² = 3. Dividing both sides by 0.5, we obtain x² = 6. Taking the square root of both sides, we find x = √6. Therefore, CM = 2√6 and MA = √6.
Now that we have the lengths of CM and MA, we can find the area of triangles CDB, AMC, ABD, ADC, and ABC using the formula for the area of a triangle.
The area of triangle CDB is given by A = 0.5 * CD * DB = 0.5 * CM * BD. Plugging in the values, we get A = 0.5 * 2√6 * 2√6 = 2 * 6 = 12 cm².
The area of triangle AMC is given by A = 0.5 * AM * CM = 0.5 * √6 * 2√6 = 2 * 6 = 12 cm².
The area of triangle ABD is given by A = 0.5 * AB * BD = 0.5 * 2 * 2√6 = 2√6 cm².
The area of triangle ADC is given by A = 0.5 * AD * DC = 0.5 * 2 * 2√ 6 = 2√6 cm².
Finally, the area of triangle ABC is given by A = 0.5 * AB * BC = 0.5 * 2 * 4√6 = 4√6 cm².