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Relation between Areas and Sides of Similar Triangles

ABC is a righ...

Question

ABC is a right angled triangle with $∠$ ABC = $90^{\circ}$ . D is any point on AB and DE is perpendicular to AC. Prove that : (i) $Δ$ ADE~ $Δ$ ACB. (ii) If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD. (iii) Find, area of $Δ$ ADE : area of quadrilateral BCED.

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Solution

$left parenthesis i i right parenthesis I n space triangle A E D space a n d space triangle A C B comma angle A E D space equals space angle A B C space.... left parenthesis b o t h space a r e space r i g h t space a n g l e s right parenthesis angle D A E space equals space angle C A B space.... left parenthesis c o m m o n space a n g l e s right parenthesis triangle A D E space tilde triangle A C B space.... left parenthesis A A space c r i t e r i o n space f o r space s i m i l a r i t y right parenthesis rightwards double arrow fraction numerator A E over denominator A B end fraction space equals space fraction numerator D E over denominator B C end fraction space equals space fraction numerator A D over denominator A C end fraction rightwards double arrow fraction numerator 4 over denominator A B end fraction space equals space fraction numerator D E over denominator 5 end fraction space equals space fraction numerator A D over denominator 13 end fraction.... left parenthesis i right parenthesis i n space r i g h t space increment A B C comma rightwards double arrow A B squared plus B C squared space equals space A C squared rightwards double arrow A B squared plus 5 squared space equals space 13 squared rightwards double arrow A B equals space 12 f r o m space left parenthesis i right parenthesis space w e space g e t space 4 over 12 space equals space fraction numerator D E over denominator 5 end fraction space equals space fraction numerator A D over denominator 13 end fraction s o comma space D E space equals space 20 over 12 space equals space 5 over 3 space equals space 1 2 over 3 c m fraction numerator A D over denominator 13 end fraction space equals space 4 over 12 rightwards double arrow A D space equals 52 over 12 space equals space 4 1 third c m$

$left parenthesis i i i right parenthesis I n space triangle A E D space a n d space triangle A C B comma angle A E D space equals space angle A B C space.... left parenthesis b o t h space a r e space r i g h t space a n g l e s right parenthesis angle D A E space equals space angle C A B space.... left parenthesis c o m m o n space a n g l e s right parenthesis triangle A D E space tilde triangle A C B space.... left parenthesis A A space c r i t e r i o n space f o r space s i m i l a r i t y right parenthesis rightwards double arrow fraction numerator A E over denominator A B end fraction space equals space fraction numerator D E over denominator B C end fraction space equals space fraction numerator A D over denominator A C end fraction rightwards double arrow fraction numerator 4 over denominator A B end fraction space equals space fraction numerator D E over denominator 5 end fraction space equals space fraction numerator A D over denominator 13 end fraction.... left parenthesis i right parenthesis i n space r i g h t space increment A B C comma rightwards double arrow A B squared plus B C squared space equals space A C squared rightwards double arrow A B squared plus 5 squared space equals space 13 squared rightwards double arrow A B equals space 12 f r o m space left parenthesis i right parenthesis space w e space g e t space 4 over 12 space equals space fraction numerator D E over denominator 5 end fraction space equals space fraction numerator A D over denominator 13 end fraction fraction numerator a r left parenthesis triangle A D E right parenthesis over denominator a r left parenthesis B C E D right parenthesis end fraction space equals space fraction numerator A E squared over denominator A B squared end fraction space equals space 4 squared over 12 squared space equals space 16 over 144 rightwards double arrow fraction numerator a r left parenthesis increment A D E right parenthesis over denominator a r left parenthesis B C E D right parenthesis end fraction space equals space 1 over 9$

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