Let X, Y and Z be the midpoints of the sides BC, CA and AB of a triangle ABC respectively. Consider the following statements:
I. The quadrilateral AZXY is a parallelogram.
II. The area of the quadrilateral AZXY is half of the area of the triangle ABC.
Which of the statements given above is/are correct?
Understanding Triangle Midpoints and Properties
This problem involves a triangle ABC and its midpoints X, Y, and Z on sides BC, CA, and AB, respectively. We need to analyze two statements about the quadrilateral AZXY formed by these points and the vertices A and Z.
The Midpoint Theorem
The Midpoint Theorem is a key concept here. It states that:
- The line segment connecting the midpoints of two sides of a triangle is parallel to the third side.
- The length of this line segment is half the length of the third side.
Applying this to triangle ABC:
- Since Z is the midpoint of AB and Y is the midpoint of AC, the line segment ZY is parallel to BC and \(ZY = \frac{1}{2} BC\).
- Similarly, XY is parallel to AB and \(XY = \frac{1}{2} AB\).
- And XZ is parallel to AC and \(XZ = \frac{1}{2} AC\).
Statement I: AZXY is a Parallelogram
Let's analyze the quadrilateral AZXY:
- Sides AZ and XY: Z is the midpoint of AB, so \(AZ = \frac{1}{2} AB\). From the midpoint theorem, we know \(XY = \frac{1}{2} AB\). Therefore, \(AZ = XY\).
- Parallelism: AZ is a part of the side AB. Since \(XY = \frac{1}{2} AB\) and XY is parallel to AB (by the midpoint theorem), it means XY is parallel to AZ.
- Conclusion for Statement I: Since one pair of opposite sides (AZ and XY) are equal in length and parallel, the quadrilateral AZXY is a parallelogram.
So, Statement I is correct.
Statement II: Area of AZXY is Half the Area of ABC
Consider the four triangles formed by joining the midpoints X, Y, and Z:
- Triangle AZY
- Triangle ZBX
- Triangle YXC
- Triangle ZYX
It's a known property that when you join the midpoints of a triangle, you divide the original triangle into four smaller triangles that are congruent to each other and similar to the original triangle.
This means:
- Area(AZY) = Area(ZBX) = Area(YXC) = Area(ZYX) = \(\frac{1}{4}\) Area(ABC)
The quadrilateral AZXY is composed of two of these smaller triangles:
- Triangle AZY
- Triangle ZYX
Therefore, the area of AZXY is:
Area(AZXY) = Area(AZY) + Area(ZYX)
Area(AZXY) = \(\frac{1}{4}\) Area(ABC) + \(\frac{1}{4}\) Area(ABC)
Area(AZXY) = \(\frac{2}{4}\) Area(ABC) = \(\frac{1}{2}\) Area(ABC)
So, Statement II is also correct.
Final Conclusion
Both Statement I (AZXY is a parallelogram) and Statement II (Area of AZXY is half the area of triangle ABC) are correct based on the properties derived from the midpoint theorem and the division of a triangle by its midsegments.
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