The line segment A B is 2 5 2 5 long and the line segment D C is perpendicular to A B . If the coordinates of the endpoint B are ( x B , y B ) , find ( x B − y B ) .
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@Ossama Ismail , please use the standard lowercase x and y and not the uppercase for axis and coordinates. Put line segments A B and D C and number 2 5 2 5 in LaTex.
Happy New Year
Wishing you a Happy New Year with the hope that you will have many blessings in the year to come. Thanks for your brilliant solutions. I am always enjoying your solutions to my problems.
The line C D can be modeled as y = − 3 4 x + 2 0 0 0 and A B as y = 4 3 x + 5 0 6 . If ∣ A B ∣ = 2 5 2 5 , then we can compute x B via the Distance Formula:
( x B − 0 ) 2 + ( y B − 5 0 6 ) 2 = ( x B − 0 ) 2 + ( 4 3 x B + 5 0 6 − 5 0 6 ) 2 = 2 5 2 5 ;
or x B 2 + 1 6 9 x B 2 = 2 5 2 5 ;
or 1 6 2 5 x B 2 = 2 5 2 5 ;
or 4 5 x B = 2 5 2 5 ;
or x B = 2 0 2 0 ⇒ y B = 4 3 ( 2 0 2 0 ) + 5 0 6 = 2 0 2 1 .
Hence, x B − y B = 2 0 2 0 − 2 0 2 1 = − 1 .
We can just solve it using similar triangles see my solution.
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Will do, Chew-Seong.....Happy 2021 and to more ingenious Brilliant solutions!!!
A B ⊥ C D ⇒ A B ⋅ C D = 0 ⇒ ( x − 0 y − 5 0 6 ) ⋅ ( 0 − 1 5 0 0 2 0 0 0 − 0 ) = 0 ⇒ − 1 5 0 0 x + 2 0 0 0 ( y − 5 0 6 ) = 0 ⇒ y − 5 0 6 = 4 3 x ( 1 )
∣ ∣ ∣ A B ∣ ∣ ∣ = 2 5 2 5 ⇒ x 2 + ( y − 5 0 6 ) 2 = 2 5 2 5 ⇒ ( 1 ) x 2 + ( 4 3 x ) 2 = 2 5 2 5 2 ⇒ x > 0 x = 2 0 2 0 ( 2 )
( 1 ) , ( 2 ) ⇒ x − y = x − 4 3 x − 5 0 6 = 4 x − 5 0 6 = 4 2 0 2 0 − 5 0 6 = − 1
Let the orange line be y = 4 3 x + 5 0 6 , the slope = 4 3 as it is perpendicular to blue line with slope = 3 − 4
x B 2 + ( y B − 5 0 6 ) 2 = 2 5 2 5 2 ⟹ x B 2 + ( 4 3 x ) 2 = 2 5 2 5 2 ⟹ x B 2 + 1 6 9 x B 2 = 2 5 2 5 2 ⟹ 2 5 x B 2 = 1 6 × 2 5 2 5 2 ⟹ 5 x B = 4 × 2 5 2 5 ⟹ x B = 4 × 5 0 5 = 2 0 2 0 ⟹ x B − y B = 4 1 x B − 5 0 6 = 5 0 5 − 5 0 6 = − 1
Of course, y B = 2 0 2 1 , Happy New Year!
Answer
= (4/5)(2525) - [(3/5)(2525) + 506]
= 2020 - 2021
= -1
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Let O be the origin ( 0 , 0 ) ; and A E and B E be parallel to the x - and y -axis respectively. Then we note that △ A B E and △ C D O are similar and △ C D O is a 3 - 4 - 5 right triangle. Since A B = 2 5 2 5 , x B = A E = 5 4 × 2 5 2 5 = 2 0 2 0 and y B = B E + 5 0 6 = 5 3 × 2 5 2 5 + 5 0 6 = 2 0 2 1 ( Happy New Year ) and x B − y B = − 1 .