Math Olympiad Test: Quadrilaterals- 1 - Class 9 MCQ

Rectangle has equal length of diagonals.

Let the angles be 3x, 5x, 9x and 13x
∴ Sum of largest and smallest angle = 3x + 13x = 16x
A/Q,
3x + 5x + 9x + 13x = 360°
⇒ 30x = 360°  ⇒ x = 12
∴ 16x = 16 × 12° = 192°

In ΔACD

ZY ∥ AC and YZ = 1/2 [Using mid-point theorem]

Rhombus is a parallelogram having consecutive sides equal and none of the angles equal a right angle.

∵ AD is the median of ΔABC
∴ BD = DC
Through D, draw DR || BF

Now, in ΔBFC,
DR || BF and D is the mid-point of BC
∴ R should be the mid-point of FC (according to converse of mid-point theorem)
∴ FR = RC ...(i)
Similarly, in ΔADR
E is the mid-point of AD and EF || DR
∴ F should be the mid-point of AR
∴ FR = AF ...(ii)
Using (i) and (ii)
FR = RC = AF
⇒ AC = 3AF
⇒ 

The resulting figure will be a square 

A kite is generally not considered a parallelogram because a kite is a quadrilateral whose four sides can be grouped into two pairs of sides of the same length that are adjacent to each other.

And also,
AB || CD, or AX || CY
∵ AX = CY and AX || CY
∴ AXCY is a parallelogram
(∴ one pair of sides are equal and parallel)

P and Q are the mid-points of BD and AC respectively

Applying mid-point theorem
PQ ∥ AB ∥ DC, and