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In a Delta AB...

Question

In a \Delta ABC, D and E are points on the sides AB and AC respectively such that DE||BC.
(i) If AD = 6cm, DB = 9 cm and AE = 8 cm, find AC.
(ii) If $\frac{A D}{D B} = \frac{3}{4}$ and AC = 15 cm, find AE.
(iii) If $\frac{A D}{D B} = \frac{2}{3}$ and AC = 18 cm, find AE.
(iv) If AD = 4, AE=8, DB =x-4, and EC = 3x - 19 find x
(v) If AD = 8 cm, AB=12 cm and AE = 12 cm, finde CE.
(vi) If ad = 4 CM, db = 4.5 cm and AE = 8 cm, find AC
(vii) If AD = 2 cm, AB = 6 cm and AC = 9cm, find AE.
(viii) If $\frac{A D}{B D} = \frac{4}{5}$ and 3C = 2.5 cm, find AE.
(ix) If AC=x, DB=x-2, AE=x+2 and EC=x-1, find the value of x.
(x) If AD= 8x -7. DB= 5x-3, AE=4x-3 and EC= (3x-1), find the value of x.
(xi) If AD= 4x-3, AE=8x -7, BC=3x-1 and CE=5x-3, find the value x.
(xii) If AD=2.5 cm, BD= =3.0 cm and AE=3.75 cm, find the length of AC.

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Solution

$left parenthesis i right parenthesis space I f space A D space equals space 6 c m comma space D B space equals space 9 space c m space a n d space A E space equals space 8 space c m sin c e space D E thin space vertical line vertical line space B C space w e space c a n space s a y space t h a t space increment A D E tilde increment A B C fraction numerator A D over denominator D B end fraction space equals space fraction numerator A E over denominator E C end fraction left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis 6 over 9 space equals space fraction numerator 8 over denominator E C end fraction rightwards double arrow E C equals space 72 over 6 space equals space 12 h e n c e space A C thin space equals space A E plus E C thin space equals space 8 plus 12 space equals space 20 c m$

$left parenthesis i i right parenthesis space I f space fraction numerator A D over denominator D B end fraction equals fraction numerator 3 space over denominator 4 end fraction a n d space A C space equals space 15 space c m fraction numerator D B over denominator A D end fraction equals fraction numerator 4 space over denominator 3 end fraction space a d d i n g space 1 space o n space b o t h space s i d e s fraction numerator D B over denominator A D end fraction plus 1 equals fraction numerator 4 space over denominator 3 end fraction plus 1 fraction numerator A D plus D B over denominator A D end fraction space equals 7 over 3 fraction numerator A B over denominator A D end fraction equals 7 over 3 bold left parenthesis bold italic u bold sin bold italic g bold space bold italic T bold italic h bold italic a bold italic l bold italic e bold italic s bold space bold italic T bold italic h bold italic e bold italic o bold italic r bold italic e bold italic m bold right parenthesis space w e space h a v e comma a n d space b y space t h e space s i m i l a r i t y space o f space A D E space a n d space A B C space w e space c a n space s a y space t h a t fraction numerator A B over denominator A D end fraction equals fraction numerator A C over denominator A E end fraction rightwards double arrow 7 over 3 equals fraction numerator 15 over denominator A E end fraction rightwards double arrow A E space equals space 45 over 7 c m$

$left parenthesis i i i right parenthesis space I f space fraction numerator A D over denominator D B end fraction equals fraction numerator 2 space over denominator 3 end fraction a n d space A C space equals space 18 space c m fraction numerator D B over denominator A D end fraction equals fraction numerator 3 space over denominator 2 end fraction space a d d i n g space 1 space o n space b o t h space s i d e s fraction numerator D B over denominator A D end fraction plus 1 equals fraction numerator 3 space over denominator 2 end fraction plus 1 fraction numerator A D plus D B over denominator A D end fraction space equals 5 over 2 fraction numerator A B over denominator A D end fraction equals 5 over 2 a n d space b y space t h e space s i m i l a r i t y space o f space A D E space a n d space A B C space w e space c a n space s a y space t h a t left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis space w e space h a v e comma fraction numerator A B over denominator A D end fraction equals fraction numerator A C over denominator A E end fraction rightwards double arrow 5 over 2 equals fraction numerator 18 over denominator A E end fraction rightwards double arrow A E space equals space 36 over 5 c m equals 7.2 c m$

$left parenthesis i v right parenthesis space I f space A D space equals space 4 comma space A E equals 8 comma space D B space equals x minus 4 comma space a n d space E C space equals space 3 x space minus space 19 fraction numerator A D over denominator D B end fraction equals fraction numerator A E over denominator E C end fraction fraction numerator up diagonal strike 4 space 1 over denominator x minus 4 end fraction equals fraction numerator up diagonal strike 8 space end strike 2 over denominator 3 x minus 19 end fraction 3 x minus 19 space equals space 2 x minus 8 h e n c e space x space equals space 11$
$left parenthesis v right parenthesis space I f space A D space equals space 8 space c m comma space A B equals 12 space c m space a n d space A E space equals space 12 space c m b y space s i m i l a r i t y space o f space t h e space t r i a n g l e space A D E thin space a n d space A B C left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis space w e space h a v e fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction rightwards double arrow 8 over 12 equals fraction numerator 12 over denominator A C end fraction rightwards double arrow A C space equals space fraction numerator 12 cross times 12 over denominator 8 end fraction equals 18 c m h e n c e space C E space equals space A C minus A E equals 18 minus 12 equals 6 c m$

$left parenthesis v i right parenthesis space I f space A D space equals space 4 space C M comma space D B space equals space 4.5 space c m space a n d space A E space equals space 8 space c m comma space f i n d space A C b y space s i m i l a r i t y space o f space t h e space t r i a n g l e space A D E thin space a n d space A B C left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis comma space w e space h a v e fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction rightwards double arrow fraction numerator A D over denominator A D plus D B end fraction equals fraction numerator A E over denominator A C end fraction rightwards double arrow fraction numerator 4 over denominator 4 plus 4.5 end fraction equals fraction numerator 8 over denominator A C end fraction rightwards double arrow A C space equals space fraction numerator 9.5 cross times 8 over denominator 4 end fraction equals 19 c m$

$left parenthesis v i i right parenthesis space I f space A D space equals space 2 space c m comma space A B space equals space 6 space c m space a n d space A C space equals space 9 c m comma space b y space s i m i l a r i t y space o f space t h e space t r i a n g l e space A D E thin space a n d space A B C left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis comma space w e space h a v e fraction numerator A D over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction rightwards double arrow 2 over 6 equals fraction numerator A E over denominator 9 end fraction rightwards double arrow A E equals fraction numerator 2 cross times 9 over denominator 6 end fraction equals 3 c m$

$left parenthesis v i i i right parenthesis space I f fraction numerator space A D over denominator B D end fraction equals 4 over 5 space a n d space E C space equals space 2.5 space c m comma space f i n d space A E. fraction numerator A D over denominator B D end fraction space equals space fraction numerator A E over denominator E C end fraction left parenthesis b y space T h a l e s space t h e o r e m right parenthesis 4 over 5 equals fraction numerator A E over denominator 2.5 end fraction rightwards double arrow A E space equals space 2 c m$

$left parenthesis i x right parenthesis space I f space A C equals x comma space D B equals x minus 2 comma space A E equals x plus 2 space a n d space E C equals x minus 1 comma space f i n d space t h e space v a l u e space o f space x W e space h a v e space t o space f i n d space t h e space v a l u e space o f space x S o comma space fraction numerator A D over denominator B D end fraction equals fraction numerator A E over denominator C E end fraction space left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis fraction numerator x over denominator x minus 2 end fraction equals fraction numerator x plus 2 over denominator x minus 1 end fraction x left parenthesis x space â�� space 1 right parenthesis space equals space left parenthesis x space â�� space 2 right parenthesis left parenthesis x space plus space 2 right parenthesis x squared space â�� space x space â�� space x squared space plus space 4 space equals space 0 x equals 4 c m$

$x right parenthesis space space I t space i s space g i v e n space t h a t space A D space equals space 8 x space â�� space 7 comma space D B space equals space 5 x space â�� space 3 comma space A E space equals space 4 x space â�� space 3 space a n d space E C space equals space 3 x space minus 1 space space W e space h a v e space t o space f i n d space t h e space v a l u e space o f space x fraction numerator A D over denominator B D end fraction equals fraction numerator A E over denominator C E end fraction left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis fraction numerator 8 x minus 7 over denominator 5 x minus 3 end fraction equals fraction numerator 4 x minus 3 over denominator 3 x minus 1 end fraction left parenthesis 8 x space â�� space 7 right parenthesis left parenthesis 3 x space â�� space 1 right parenthesis space equals space left parenthesis 5 x space â�� space 3 right parenthesis left parenthesis 4 x space â�� space 3 right parenthesis 24 x squared space â�� space 29 x space plus space 7 space equals space 20 x squared space â�� space 27 x space plus space 9 4 x squared â�� space 2 x space â�� space 2 space equals space 0 2 left parenthesis 2 x squared space â�� space x space â�� space 1 right parenthesis space equals space 0 left parenthesis 2 x squared space â�� space x space â�� space 1 right parenthesis space equals space 0 2 x squared space â�� space 2 x space plus space x space â�� space 1 space equals space 0 2 x left parenthesis x space â�� space 1 right parenthesis space plus space 1 left parenthesis x space â�� space 1 right parenthesis space equals space 0 left parenthesis x space â�� space 1 right parenthesis left parenthesis 2 x space plus space 1 right parenthesis space equals space 0 space space x space equals space 1 space o r space x space equals space minus 1 divided by 2 space space S i n c e space t h e space s i d e space o f space t r i a n g l e space c a n space n e v e r space b e space n e g a t i v e space space T h e r e f o r e comma space x space equals space 1.$

$left parenthesis x i right parenthesis I t space i s space g i v e n space t h a t space A D space equals space 4 x space â�� space 3 comma space B D space equals space 3 x space â�� space 1 comma space A E space equals space 8 x space â�� space 7 space a n d space E C space equals space 5 x space â�� space 3 space F o r space f i n d i n g space t h e space v a l u e space o f space x. fraction numerator A D over denominator B D end fraction equals fraction numerator A E over denominator E C end fraction left parenthesis b y space T h a l e s space T h e o r e m right parenthesis fraction numerator 4 x minus 3 over denominator 3 x minus 1 end fraction equals fraction numerator 8 x minus 7 over denominator 5 x minus 3 end fraction left parenthesis 4 x space â�� space 3 right parenthesis left parenthesis 5 x space â�� space 3 right parenthesis space equals space left parenthesis 3 x space â�� space 1 right parenthesis left parenthesis 8 x space â�� space 7 right parenthesis 4 x left parenthesis 5 x space â�� space 3 right parenthesis space minus 3 left parenthesis 5 x space â�� space 3 right parenthesis space equals space 3 x left parenthesis 8 x space â�� space 7 right parenthesis space minus 1 left parenthesis 8 x space â�� space 7 right parenthesis 20 x squared space â�� space 12 x space â�� space 15 x space plus space 9 space equals space 24 x squared space â�� space 29 x space plus space 7 20 x squared space minus 27 x space plus space 9 space equals space 24 x squared space minus 29 x space plus space 7 minus 4 x squared space plus space 2 x space plus space 2 space equals space 0 4 x squared space â�� space 2 x space â�� space 2 space equals space 0 left parenthesis 2 x squared space â�� space x space â�� space 1 right parenthesis space equals space 0 2 x squared space â�� space 2 x space plus space x space â�� space 1 space equals space 0 2 x left parenthesis x space â�� space 1 right parenthesis space plus space 1 left parenthesis x space â�� space 1 right parenthesis space equals space 0 left parenthesis x space â�� space 1 right parenthesis left parenthesis 2 x space plus space 1 right parenthesis space equals space 0 space space x space equals space 1 space o r space x space equals space minus 1 divided by 2 space space S i n c e space t h e space s i d e space o f space t r i a n g l e space c a n space n e v e r space b e space n e g a t i v e space space T h e r e f o r e comma space x space equals space 1.$

$left parenthesis x i i right parenthesis space I f space A D space equals space 2.5 c m comma space D B space equals space 3 space c m space a n d space A E space equals space 3.75 space c m sin c e space D E thin space vertical line vertical line space B C space w e space c a n space s a y space t h a t space increment A D E tilde increment A B C fraction numerator A D over denominator D B end fraction space equals space fraction numerator A E over denominator E C end fraction left parenthesis u sin g space T h a l e s space T h e o r e m right parenthesis fraction numerator 2.5 over denominator 3 end fraction space equals space fraction numerator 3.75 over denominator E C end fraction rightwards double arrow E C equals space fraction numerator 3.75 cross times 3 over denominator 2.5 end fraction space equals space fraction numerator 3 cross times 3 over denominator 2 end fraction equals 4.5 h e n c e space A C thin space equals space A E plus E C thin space equals space 4.5 plus 3.75 space equals space 8.25 c m$

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Similar questions

Q. In a ∆ABC, D and E are points on the sides AB and AC respectively such that DE || BC.

(i) If AD = 6 cm, DB = 9 cm and AE = 8 cm, find AC.
(ii) If

\frac{AD}{DB} = \frac{3}{4}

and AC = 15 cm, find AE.
(iii) If

\frac{AD}{DB} = \frac{2}{3}

and AC = 18 cm, find AE.
(iv) If AD = 4, AE = 8, DB = x − 4, and EC = 3x − 19, find x.
(v) If AD = 8 cm, AB = 12 cm and AE = 12 cm, find CE.
(vi) if AD = 4 cm, DB = 4.5 cm and AE = 8 cm, find AC.
(vii) If AD = 2 cm, AB = 6 cm, and AC = 9 cm, find AE.
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Q. In the given figure, DE

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D and E are points on the sides AB and AC respectively of a $Δ A B C$ such that DE || BC
Find the value of x, when

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