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Equation of a Line Perpendicular to a Given Line

Show that A 3...

Question

Show that A (3, 2), B (6, -2) and C (2, -5) can be the vertices of a square.

(i) Find the co-ordinates of its fourth vertex D, if ABCD is a square.

(ii) Without using the co-ordinates of vertex D, find the equation of side AD of the square and also the equation of diagonal BD.

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Solution

A (3, 2), B (6, -2) and C (2, -5) $A space left parenthesis 3 comma space 2 right parenthesis comma space B space left parenthesis 6 comma space minus 2 right parenthesis space a n d space C space left parenthesis 2 comma space minus 5 right parenthesis f o r space b e i n g space a space s q u a r e space A B space s h o u l d space b e space p e r p e n d i c u l a r space t o space A C a n d space open vertical bar A B close vertical bar space equals space open vertical bar A C close vertical bar s l o p e space o f space A B space equals space fraction numerator B subscript y minus A subscript y over denominator B subscript x minus A subscript x end fraction space equals space fraction numerator negative 2 minus 2 over denominator 6 minus 3 end fraction space equals space fraction numerator negative 4 over denominator 3 end fraction s l o p e space o f space B C space equals space fraction numerator C subscript y minus B subscript y over denominator C subscript x minus B subscript x end fraction space equals space fraction numerator negative 5 minus left parenthesis negative 2 right parenthesis over denominator 2 minus 6 end fraction space equals space fraction numerator negative 3 over denominator negative 4 end fraction space equals space 3 over 4 h e n c e space t h e space p r o d u c t space o f space s l o p e s space i s space 3 over 4 cross times open parentheses fraction numerator negative 4 over denominator 3 end fraction close parentheses space equals space minus 1 h e n c e space t h e space l i n e s space a r e space p e r p e n d i c u l a r open vertical bar A B close vertical bar space equals space square root of open parentheses B subscript y minus A subscript y close parentheses squared plus open parentheses B subscript x minus A subscript x close parentheses squared end root space equals space square root of open parentheses negative 4 close parentheses squared plus open parentheses 3 close parentheses squared end root space equals space 5 open vertical bar B C close vertical bar space equals space square root of open parentheses C subscript y minus B subscript y close parentheses squared plus open parentheses C subscript x minus B subscript x close parentheses squared end root space equals square root of open parentheses negative 4 close parentheses squared plus open parentheses negative 3 close parentheses squared end root space equals space 5 h e n c e space t h e y space c a n space b e space t h e space v e r i t c e s space o f space a space s q u a r e s o t h e space f o r space D space w e space n e e d space t o space f i n d space t h e space e q u a t i o n space o f space A D space a n d space C D space a n d space f i n d space t h e i r space p o i n t space o f space i n t e r s e c t i o n space w h i c h space w i l l space b e space D s l o p e space o f space A D thin space equals space s l o p e space o f space B C space a s space t h e y space s h o u l d space b e space p a r a l l e l s o space e q u a t i o n space o f space A D identical to y space equals space m x plus c$
$y space equals space 3 over 4 x plus c t h i s space l i n e space s h o u l d space c r o s s space t h r o u g h space A A space equals space open parentheses 3 comma space 2 close parentheses 2 space equals space 3 over 4 open parentheses 3 close parentheses plus c c space equals space 2 minus 9 over 4 space equals space minus 1 fourth h e n c e y space equals space 3 over 4 x minus 1 fourth s o space 4 y space equals space 3 x minus 1..... left parenthesis 1 right parenthesis a n d f o r space e q u a t i o n space o f space C D s l o p e space o f space C D thin space equals space s l o p e space o f space A B s l o p e space o f space C D thin space equals space minus 4 over 3 y space equals space minus 4 over 3 x plus c t h i s space l i n e space p a s s e s space t h r o u g h space C open parentheses 2 comma space minus 5 close parentheses s o space minus 5 space equals space minus 4 over 3 left parenthesis 2 right parenthesis space plus c minus 5 space plus 8 over 3 space equals space c c space equals space minus 7 over 3 s o space y space equals space minus 4 over 3 x minus 7 over 3 3 y space equals space minus 4 x minus 7... left parenthesis 2 right parenthesis i n t e r s e c t i o n space o f space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis 4 y space equals space 3 x minus 1... left parenthesis 1 right parenthesis 3 y space equals space minus 4 x minus 7... left parenthesis 2 right parenthesis f r o m space e q u a t i o n space left parenthesis 1 right parenthesis space y space equals space fraction numerator 3 x minus 1 over denominator 4 end fraction 3 open parentheses fraction numerator 3 x minus 1 over denominator 4 end fraction close parentheses space equals space minus 4 x minus 7 9 x minus 3 space equals space minus 16 x minus 28 25 x space equals space minus 25 x space equals space minus 1 a n d space y space equals space fraction numerator 3 open parentheses negative 1 close parentheses minus 1 over denominator 4 end fraction space equals space minus 1 s o D identical to open parentheses negative 1 comma space minus 1 close parentheses$

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Similar questions

A

(- 2, 4), C (4, 10)

D (- 2, 10)

are the vertices of a square

A B C D

. Use the graphical method to find the co-ordinates of the fourth vertex

B

.

The co-ordinates of the mid-point of BC is

(m, 7)

m

The co-ordinates of the three vertices of a rectangle

A B C D

are given. By plotting the given points, find the co-ordinates of the fourth vertex:

A (4, 2), B (- 2, 2)

D (4, - 2)

.

A (- 2, 2), B (8, 2)

C (4, - 4)

are the vertices of a parallelogram

A B C D

. By plotting the given points on a graph paper; find the co-ordinates of the fourth vertex

D

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Also, from the same graph, state the co-ordinates of the mid-points of the sides

A B

C D

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Q. The extremities of the diagonal of a square are the points (1,5) and (8 , 8). Find the equation to its sides and the co-ordinates of the other vertices.

ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, -4). Find :

(i) co-ordinates of A

(ii) equation of diagonal BD.

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