Math Olympiad Test: Quadrilaterals- 1 - Free MCQ with solutions Class 9

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°

​Given: 

• ABCD is a parallelogram.
• W, X, Y, Z are midpoints of sides AB, BC, CD, and DA respectively.
• AC is a diagonal.

We are asked about YZ in relation to AC.

Using Mid Point Theorem:
The line joining the midpoints of two sides of a triangle is parallel to the third side and half of it.

In ΔADC,
Z and Y are midpoints of AD and DC respectively.
By the midpoint theorem, YZ is parallel to AC 
So,

Correct Option: B

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