Question

The length of the two parallel sides of a trapezium
are 29 cm and 11 cm and the length of each of the
non-parallel sides is 41 cm, find the area of the
trapezium.​

Answer 1

$\large\underline{\sf{Given- }}$

• The length of the two parallel sides of a trapezium are 29 cm and 11 cm.

• The length of each of the non-parallel sides is 41 cm.

$\large\underline{\sf{To\:Find - }}$

• The area of the trapezium.

$\large\underline{\sf{Solution-}}$

Let us assume that ABCD be a trapezium such that

• AB || CD

• AB = 11 cm

• BC = 41 cm

• CD = 29 cm

• DA = 41 cm

Construction :-

• Draw AL and BM perpendicular to CD intersecting CD at L and M respectively.

Now,

$\bf :\longmapsto\:In \: \triangle \: ALD \: and \: \triangle \: BMC$

$\rm :\longmapsto\:AD \: = \: BC \: \: \: \: \: \{given \}$

$\rm :\longmapsto\:AL \: = \: BM \: \: \{distance \: between \: \parallel \: lines \}$

$\rm :\longmapsto\: \angle \: ALD = \angle \: BMC \: \: \{each \: 90 \degree \}$

$\bf :\longmapsto\: \triangle ALD \: \cong \: \triangle BMC \: \{ \: RHS \: Congruency\}$

$\rm :\implies\:LD \: = \: MC \: = \: x$

Now,

• It is given that

$\rm :\longmapsto\:CD \: = \: 29$

$\rm :\longmapsto\:DL + LM + MC = 29$

$\rm :\longmapsto\:x + 11 + x = 29$

$\rm :\longmapsto\:2x = 29 - 11$

$\rm :\longmapsto\:2x = 18$

$\rm :\implies\:x = 9 \: cm$

Now,

$\rm :\longmapsto\:In \: \triangle \: ALD$

$\rm :\longmapsto\:Using \: Pythagoras \: Theorem$

$\rm :\longmapsto\: {AL}^{2} + {LD}^{2} = {AD}^{2}$

$\rm :\longmapsto\: {AL}^{2} + {(9)}^{2} = {(41)}^{2}$

$\rm :\longmapsto\: {AL}^{2} + 81 = 1681$

$\rm :\longmapsto\: {AL}^{2} = 1600$

$\rm :\longmapsto\:AL = \sqrt{1600}$

$\bf\implies \:AL = 40 \: cm$

Therefore,

$\rm :\longmapsto\:Area_{(trapezium)} = \dfrac{1}{2} (sum \: of \parallel \: sides) \times distance$

$\rm :\longmapsto\:Area_{(trapezium)} = \dfrac{1}{2} \times (AB + CD) \times AL$

$\rm :\longmapsto\:Area_{(trapezium)} = \dfrac{1}{2} \times (11 + 29) \times 40$

$\rm :\longmapsto\:Area_{(trapezium)} = 20 \times 40$

$\bf :\longmapsto\:Area_{(trapezium)} = 800 \: {cm}^{2}$