The midpoints of sides AB, BC, and AC in △ ABC are M, N, and P, respectively. The medians drawn from A, B, and C intersect the line segments MP, MN and NP at X, Y, and Z, respectively. If the area of △ ABC is 1440 sq cm, then the area, in sq cm, of △ XYZ is

Question

Accepted Answer

The given problem involves calculating the area of triangle XYZ formed by the intersection of medians in triangle ABC. The area of triangle ABC is provided as 1440 sq cm.

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### Step 1: Area of triangle MNP  
Using the **Midpoint Theorem**, the triangle formed by joining the midpoints of the sides of a triangle (MNP) has an area equal to **one-fourth** the area of the original triangle (ABC).

Hence:  
$$
	ext{Area of } 	riangle MNP = \frac{1}{4} 	imes 	ext{Area of } 	riangle ABC
$$  
Substituting values:  
$$
	ext{Area of } 	riangle MNP = \frac{1}{4} 	imes 1440 = 360 \, 	ext{sq cm}
$$

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### Step 2: Area of triangle XYZ  
Triangle XYZ is formed by the intersection points of medians inside triangle MNP. Again, using the **Midpoint Theorem**, the area of triangle XYZ is **one-fourth** the area of triangle MNP.

Hence:  
$$
	ext{Area of } 	riangle XYZ = \frac{1}{4} 	imes 	ext{Area of } 	riangle MNP
$$  
Substituting values:  
$$
	ext{Area of } 	riangle XYZ = \frac{1}{4} 	imes 360 = 90 \, 	ext{sq cm}
$$

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### Final Answer:
The area of triangle XYZ is **90 sq cm**.