Tag: trapeziums and kites

Questions Related to trapeziums and kites

The line joining the mid points of the diagonals of a trapezium has length $3$cm. If the longer base is $97$cm then the shorter base is:

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. $94$cm

  2. $92$cm

  3. $91$cm

  4. $90$cm

Show answer
Correct Option: C
Explanation:

The line joining the mid point of the diagonals of a trapezium is half the length of the difference between the two sides.
Let the smaller side be $x$
Then, $3 = \dfrac{97 -x}{2}$
$6= 97 - x$
$x = 91$ cm

The consecutive angles of a trapezium form an arithmetic sequence. If the smallest angle is $\displaystyle 75^{\circ}$, then the largest angle is

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. $\displaystyle 100^{\circ}$

  2. $\displaystyle 105^{\circ}$

  3. $\displaystyle 110^{\circ}$

  4. $\displaystyle 115^{\circ}$

Show answer
Correct Option: B
Explanation:

Since, sum of all the four angles of a quadrilateral is $360^o$.

Angle 1 $=75^o$, Angle 2 $=75^o+x$, Angle 3 $=75^o+2x$, Angle 4 $=75^o+3x$
Angle 1 $+$ Angle 2 $+$ Angle 3 $+$ Angle 4 $=360^o$
$\therefore   75^o+75^o+x+75^o+2x+75^o+3x=360^o$
$\Rightarrow 300+6x=360\Rightarrow 6x=60 \Rightarrow x=10$
$\therefore$ Largest angle (Angle 4)$=75^o+3x=75^o+3\times 10=105^o$

Hence, option B.

In a trapezium. ABCD, $\angle ADC = 110^o$. Find $\angle A$.

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. $50^o$

  2. $60^o$

  3. $70^o$

  4. $80^o$

Show answer
Correct Option: C
Explanation:

In a trapezium,     AB || CD
$\therefore$ sum of adjacent angles $= 180^o$
$\angle D + \angle A = 180^o$
$\angle A = 180^o - \angle D = 180^o - 110^o$
$\angle A= 70^o$

In isoceles trapezoid ABCD, side CD is parallel to to side AB, line segment AC is congruent to line segment BD.The degree measure of angle BDC = $80^o$. Find the measures of the $\angle A$.

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. $90^o$

  2. $100^o$

  3. $110^o$

  4. $120^o$

Show answer
Correct Option: B
Explanation:

As per the property of isosceles trapezoid, 
Opposite sides of an isosceles trapezoid are the same length (congruent) and the angles on either side of the bases are the same size (congruent).
So, Angle C = $80^o$
Since the top and bottom angles are supplementary, we know that,
Angle A = $180 - 80$
Angle A = $100^o$
Similarly, the Angle of B = $100^o$

State whether true or false:

All trapeziums are parallelograms.

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms

In the trapezium PQRS, PQ is  parallel to RS and the diagonals intersect at O. If $OP.SR= m(OR . PQ)$, then the value of m is :

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. $\dfrac{1}{4}$

  2. $\dfrac{1}{3}$

  3. $1$

  4. $\dfrac{1}{2}$

Show answer
Correct Option: C
Explanation:

In Trapezium PQRS, $\Delta OPQ$ is similar to $\Delta OSR$ by AA similarity.
$\therefore \dfrac{OP}{OR}=\dfrac{OS}{OQ}=\dfrac{PQ}{SR}$
$\therefore OP.SR=OR.PQ$
Hence, $m=1$

The parallel sides of a trapezium are $x$ and $y$ in length. The length of the line segment joining the mid points of the non parallel sides is:

maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

  1. $\dfrac{x+y}{2}$

  2. $x+y$

  3. $\dfrac{2x+3y}{2}$

  4. $\dfrac{xy}{2}$

Show answer
Correct Option: A
Explanation:

The line segment joining the mid points of non parallel sides of a trapezium is the average of sum of the parallel sides.
Hence, $= \dfrac{x+y}{2}$