Question

Find LW from the given figure ?

Hope my writing is easily Understandable :) ​

Answer 1

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

In right triangle ALY, right angled at A

By using Pythagoras Theorem,

$\rm :\longmapsto\: \boxed{\tt{ {LY}^{2} = {AL}^{2} + {AY}^{2}}} - - - (1)$

Now, In right triangle WAY, right angled at A

By using Pythagoras Theorem,

$\rm :\longmapsto\:\boxed{\tt{ {WY}^{2} = {WA}^{2} + {AY}^{2}}} - - - - (2)$

Now, In right triangle WLY, right angled at Y

By using Pythagoras Theorem,

$\rm :\longmapsto\: {WL}^{2} = {WY}^{2} + {LY}^{2}$

$\rm :\longmapsto\: {(AL + WA)}^{2} = {WY}^{2} + {LY}^{2}$

$\rm :\longmapsto\: {AL}^{2} + {WA}^{2} + 2AL \times WA = {WY}^{2} + {LY}^{2}$

$\rm :\longmapsto\: 2(AL)(WA) = - {AL}^{2} - {WA}^{2} + {WY}^{2} + {LY}^{2}$

$\rm :\longmapsto\: 2(AL)(WA) = ({WY}^{2} - {WA}^{2} ) + ( {LY}^{2} - {AL}^{2})$

So, using equation (1) and (2), we get

$\rm :\longmapsto\: 2(AL)(WA) = {AY}^{2} + {AY}^{2}$

$\rm :\longmapsto\: (AL)(WA) = {AY}^{2}$

On substituting the values of AY and AL, we get

$\rm :\longmapsto\: 3.5 \times WA= {7}^{2}$

$\rm :\longmapsto\:WA = \dfrac{49}{3.5}$

$\bf\implies \:WA = 14 \: units$

Thus,

$\rm :\longmapsto\:LW$

$\rm \:  =  \: AL + AW$

$\rm \:  =  \: 3.5 + 14$

$\rm \:  =  \: 17.5 \: units$

$\\ \purple{\rm\implies \:\:\boxed{ \: \: \bf{ LW \: = \: 17.5 \: units \: }}}$

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Additional information

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem : -

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.

Answer 2

Answer:

In right triangle ALY, right angled at A

By using Pythagoras Theorem,

\rm :\longmapsto\: \boxed{\tt{ {LY}^{2} = {AL}^{2} + {AY}^{2}}} - - - (1):⟼

LY

2

=AL

2

+AY

2

−−−(1)

Now, In right triangle WAY, right angled at A

By using Pythagoras Theorem,

\rm :\longmapsto\:\boxed{\tt{ {WY}^{2} = {WA}^{2} + {AY}^{2}}} - - - - (2):⟼

WY

2

=WA

2

+AY

2

−−−−(2)

Now, In right triangle WLY, right angled at Y

By using Pythagoras Theorem,

\rm :\longmapsto\: {WL}^{2} = {WY}^{2} + {LY}^{2}:⟼WL

2

=WY

2

+LY

2

\rm :\longmapsto\: {(AL + WA)}^{2} = {WY}^{2} + {LY}^{2}:⟼(AL+WA)

2

=WY

2

+LY

2

\rm :\longmapsto\: {AL}^{2} + {WA}^{2} + 2AL \times WA = {WY}^{2} + {LY}^{2}:⟼AL

2

+WA

2

+2AL×WA=WY

2

+LY

2

\rm :\longmapsto\: 2(AL)(WA) = - {AL}^{2} - {WA}^{2} + {WY}^{2} + {LY}^{2}:⟼2(AL)(WA)=−AL

2

−WA

2

+WY

2

+LY

2

\rm :\longmapsto\: 2(AL)(WA) = ({WY}^{2} - {WA}^{2} ) + ( {LY}^{2} - {AL}^{2}):⟼2(AL)(WA)=(WY

2

−WA

2

)+(LY

2

−AL

2

)

So, using equation (1) and (2), we get

\rm :\longmapsto\: 2(AL)(WA) = {AY}^{2} + {AY}^{2}:⟼2(AL)(WA)=AY

2

+AY

2

\rm :\longmapsto\: (AL)(WA) = {AY}^{2}:⟼(AL)(WA)=AY

2

On substituting the values of AY and AL, we get

\rm :\longmapsto\: 3.5 \times WA= {7}^{2}:⟼3.5×WA=7

2

\rm :\longmapsto\:WA = \dfrac{49}{3.5}:⟼WA=

3.5

49

\bf\implies \:WA = 14 \: units⟹WA=14units

Thus,

\rm :\longmapsto\:LW:⟼LW

\rm \: = \: AL + AW = AL+AW

\rm \: = \: 3.5 + 14 = 3.5+14

\rm \: = \: 17.5 \: units = 17.5units

\begin{gathered} \\ \purple{\rm\implies \:\:\boxed{ \: \: \bf{ LW \: = \: 17.5 \: units \: }}} \\ \end{gathered}

⟹

LW=17.5units

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

Additional information

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem : -

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.