1) x and y are respectively the mid points of sides AB and BC of a parallelogram ABCD DX and - Maths - Understanding Quadrilaterals

Question

1) x and y are respectively the mid points of sides AB and BC of
a parallelogram ABCD DX and DY intersect AC at M and N respectively
If AC = 4 5 ,find MN
2) The angle between two altitudes of a parallelogram through
the same vertex of an obtuse angle of the parallelogram is 60
degree Find the angles of the parallelogram
plzzz answer this question sir

Lalit Mehra · Accepted Answer

ABCD is parallelogram X and Y are the mid point of the sides AB and BC respectively DX and DY intersect AC in M and N respectively In Î”ABC, X and Y are the mid point of AB and BC respectively $∴ XY = \frac{1}{2} AC and XY || AC (1) (Mid point theorem)$ In Î”AOB, X is the mid point of AB and XS || OM (XY || AC) âˆ´ S is the mid point of OB (Converse of mid point theorem) â‡’ OS = SB We know that, diagonals of the parallelogram bisect each other âˆ´ OD = OB â‡’ OD = OS + SB â‡’ OD = 2OS (OS = SB) $\Rightarrow \frac{OD}{OS} = 2 \Rightarrow \frac{OS}{OD} = \frac{1}{2} \Rightarrow \frac{OS}{OD} + 1 = \frac{1}{2} + 1 \Rightarrow \frac{OS + OD}{OD} = \frac{3}{2} \Rightarrow \frac{DS}{OD} = \frac{3}{2} \Rightarrow \frac{OD}{DS} = \frac{2}{3}$ Î”DMO âˆ¼ Î”DXP (AA Similarity) $∴ \frac{MO}{XS} = \frac{OD}{DS} (Corresponding sides are proportional) \Rightarrow \frac{MO}{XS} = \frac{2}{3} \Rightarrow MO= \frac{2}{3} X S (2) Similarly, ON = \frac{2}{3} S Y (3)$ Adding (2) and (3), we get $MO + ON = \frac{2}{3} (XS +SY) ∴ MN = \frac{2}{3} XY \Rightarrow MN = \frac{2}{3} \times \frac{1}{2} AC [Using (1)] \Rightarrow MN= \frac{1}{3} \times 4 5 cm = 1 5 cm$ 2

ABCD is a parallelogram AX âŠ¥ BC, AY âŠ¥ CD and âˆ XAY = 60Â° AXCY is a quadrilateral âˆ´ âˆ XAY + âˆ AYC + âˆ C + âˆ AXC = 360Â° â‡’ 60Â° + 90Â° + âˆ C + 90Â° = 360Â° â‡’ âˆ C = 360Â° â€“ 240Â° = 120Â° âˆ A = âˆ C = 120Â° (Opposite angles of parallelogram are equal) Now, BC || AD (Opposite sides of the parallelogram are parallel) âˆ´ âˆ C + âˆ D = 180Â° (If two parallel are intersected by a transversal, then the sum of adjacent interior angles is 180Â°) â‡’ 120Â° + âˆ D = 180Â° â‡’ âˆ D = 180Â° â€“ 120Â° = 60Â° âˆ B = âˆ D = 60Â° (Opposite angles of parallelogram are equal)