Answer: The area of triangle CDB is 3 cm². The area of triangle AMC is 6 cm². The area of triangle ABD is 6 cm². The area of triangle ADC is 3 cm². The area of triangle ABC is 12 cm².
Explanation:
It is given that in △ABC, point M is the midpoint of AB , point D is the midpoint of CM , and area of BMD=3 cm².
The median is divide the area of the triangle in two equal parts.
Since M is the midpoint of AB, therefore MC is the median and the area of triangle AMC and area of triangle BMC are equal.
[tex]\text{Area of } \triangle AMC=\text{Area of } \triangle BMC[/tex] .... (1)
It is given that the point D is the midpoint of CM, therefore BD is the median and the area of BMD is equal to the area of CBD.
[tex]\text{Area of } \triangle CBD=\text{Area of } \triangle BMD[/tex]
Since the area of BMD is 3 cm².
[tex]\text{Area of } \triangle CBD=3[/tex]
Therefore, the area of triangle CBD is 3 cm².
[tex]\text{Area of } \triangle BMC=\text{Area of } \triangle BMD+\text{Area of } \triangle CBD[/tex]
[tex]\text{Area of } \triangle BMC=3+3[/tex]
[tex]\text{Area of } \triangle BMC=6[/tex]
From equation (1).
[tex]\text{Area of } \triangle AMC=6[/tex]
Therefore, the area of triangle AMC is 6 cm².
It is given that the point D is the midpoint of CM, therefore AD is the median and the area of AMD is equal to the area of ADC.
[tex]\text{Area of } \triangle AMD=\text{Area of } \triangle ADC[/tex] .... (2)
Since M is the midpoint of AB therefore area of AMD and BMD are same and the area of BMD is 3 cm².
[tex]\text{Area of } \triangle ADC=3[/tex].
Therefore the area of triangle ADC is 3 cm².
[tex]\text{Area of } \triangle ABD=\text{Area of } \triangle AMD+\text{Area of } \triangle BMD[/tex]
[tex]\text{Area of } \triangle ABD=3+3=6[/tex]
Therefore the area of triangle ABD is 6 cm².
[tex]\text{Area of } \triangle ABC=2\times \text{Area of } \triangle AMC[/tex]
[tex]\text{Area of } \triangle ABC=2\times 6[/tex]
[tex]\text{Area of } \triangle ABC=12[/tex]
Therefore the area of triangle ABC is 12 cm².
