Question

the diagonal of a rectangular is 60m more than the smaller side of the rectangular. if the longer side is 30m more than the smaller side, find the sides of the playground​

Answer 1

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

Let us assume

• A rectangle ABCD

such that

• AB is longer side.

• BC is smaller side.

• AC is diagonal.

According to statement,

• The diagonal of a rectangular is 60m more than the smaller side of the rectangular and the longer side is 30m more than the smaller side.

Let

• Smaller side, BC = x meter.

So,

• Longer side, BC = (x + 30) meter

and

• Diagonal, AC = (x + 60) meter

Now,

$\rm :\longmapsto\:In \: right \: \triangle \: ABC,$

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

$\rm :\longmapsto\: {AB}^{2} + {BC}^{2} = {AC}^{2}$

$\rm :\longmapsto\: {x}^{2} + {(x + 30)}^{2} = {(x + 60)}^{2}$

$\rm :\longmapsto\: \cancel{{x}^{2}} + {x}^{2} + 900 + 60x = \cancel{{x}^{2}} + 3600 + 120x$

$\rm :\longmapsto\: {x}^{2} - 60x - 2700 = 0$

$\rm :\longmapsto\: {x}^{2} - 90x + 30x - 2700 = 0$

$\rm :\longmapsto\:x(x - 90) + 30(x - 90) = 0$

$\rm :\longmapsto\:(x - 90)(x + 30) = 0$

$\rm :\implies\:x = 90 \: \: \: or \: \: \: x = - \: 30 \: \: \: \: \{ \red{ \sf \: rejected} \}$

So,

Dimensions are,

• Smaller side, BC = 90 meter

and

• Longer side, AB = 90 + 30 = 120 meter.

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac

Answer 2

Step-by-step explanation:

Solution−

Let us assume

A rectangle ABCD

such that

AB is longer side.

BC is smaller side.

AC is diagonal.

According to statement,

The diagonal of a rectangular is 60m more than the smaller side of the rectangular and the longer side is 30m more than the smaller side.

Let

Smaller side, BC = x meter.

So,

Longer side, BC = (x + 30) meter

and

Diagonal, AC = (x + 60) meter

Now,

\rm :\longmapsto\:In \: right \: \triangle \: ABC,:⟼Inright△ABC,

\rm :\longmapsto\:Using \: Pythagoras \: Theorem:⟼UsingPythagorasTheorem

\rm :\longmapsto\: {AB}^{2} + {BC}^{2} = {AC}^{2}:⟼AB

2

+BC

2

=AC

2

\rm :\longmapsto\: {x}^{2} + {(x + 30)}^{2} = {(x + 60)}^{2}:⟼x

2

+(x+30)

2

=(x+60)

2

\rm :\longmapsto\: \cancel{{x}^{2}} + {x}^{2} + 900 + 60x = \cancel{{x}^{2}} + 3600 + 120x:⟼

x

2

+x

2

+900+60x=

x

2

+3600+120x

\rm :\longmapsto\: {x}^{2} - 60x - 2700 = 0:⟼x

2

−60x−2700=0

\rm :\longmapsto\: {x}^{2} - 90x + 30x - 2700 = 0:⟼x

2

−90x+30x−2700=0

\rm :\longmapsto\:x(x - 90) + 30(x - 90) = 0:⟼x(x−90)+30(x−90)=0

\rm :\longmapsto\:(x - 90)(x + 30) = 0:⟼(x−90)(x+30)=0

\rm :\implies\:x = 90 \: \: \: or \: \: \: x = - \: 30 \: \: \: \: \{ \red{ \sf \: rejected} \}:⟹x=90orx=−30{rejected}

So,

Dimensions are,

Smaller side, BC = 90 meter

and

Longer side, AB = 90 + 30 = 120 meter.

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac