UPSC CDS II 2025 Mathematics Paper

Concept:The midpoints of the sides of a triangle create parallel and equal segments, forming a parallelogram. The area of such a parallelogram is half the area of the original triangle.Explanation:X, Y, Z are midpoints of BC, CA, AB respectively.By the midpoint theorem, $ZY \parallel BC$ and $ZY = \frac{1}{2}BC$. Also, $XY \parallel AB$ and $XY = \frac{1}{2}AB$, and $ZX \parallel AC$ and $ZX = \frac{1}{2}AC$.In quadrilateral AZXY: side $AZ$ lies on $AB$ and is half of $AB$, while side $XY$ is parallel to $AB$ and also half of $AB$. Thus $AZ \parallel XY$ and $AZ = XY$. Similarly, $AY \parallel ZX$ and $AY = ZX$. Hence AZXY is a parallelogram. So statement I is correct.Area of parallelogram AZXY = base $AZ \times$ height. Base $AZ = \frac{1}{2}AB$. Height from $Y$ to line $AZ$ is half the altitude from $C$ to $AB$, i.e., $\frac{h}{2}$. So area = $\frac{1}{2}AB \times \frac{h}{2} = \frac{AB \cdot h}{4}$. Since area of $\triangle ABC = \frac{1}{2}AB \cdot h$, the parallelogram area equals half of the triangle's area. Thus statement II is also correct.Answer:Both I and II are correct. Option C.