O
ordey
what is the
a 2 b 4
5
3
Answers
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11th
Maths
Straight Lines
Introduction
Show that A(2, 3), B(4, 5) ...
MATHS
Show that A(2,3),B(4,5) and C(3,2) can be the vertices of a rectangle. Find the coordinates of the fourth vertex.
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ANSWER
Given: A=(2,3);B=(4,5);C=(3,2)
Slope of AB=
4−2
5−3
=1
Slope of BC=
3−4
2−5
=
−1
−3
=3
Slope of AC=
3−2
2−3
=−1
Slope of AB× Slope of AC=−1
∴AB⊥AC
Let D=(x,y)
Slope of AC=Slope of BD
∴
2−3
3−2
=
4−x
5−y
⇒5−y=x−4⇒x+y−9=0
Also, Slope of BD× Slope of DC=−1
⇒
4−x
5−y
×
x−3
y−2
=−1
⇒(5−y)(y−2)=(3−x)(4−x)
⇒5y−10−y
2
+2y=12−3x−4x+x
2
⇒y
2
−7y+10=−x
2
+7x−12
Puttig y=9−x
(9−x)
2
−7(9−x)+10+x
2
−7x+12=0
⇒81+x
2
−18x−63+7x+10+x
2
−7x+10=0
⇒2x
2
−18x+40=0
x
2
−9x+20=0⇒x=5 or x=4
If x=5,y=4
If x=4,y=5
(4,5) is already a vertex (B)
∴D=(5,4)
∴Four vertices are A(2,3);B(4,5);C(3,2);D(5,4)
I hope this can help you