Answer to Question #299234 in Analytic Geometry for Philly

Question #299234

In the Cartesian plane, two vertices of a square have coordinates (3 ; 4) and ( -2 ; - 1). One of the other two vertices, has coordibates


1
Expert's answer
2022-02-22T10:45:00-0500

Two opposite vertices of a square are (3,4)(3,4) and (2,1).(-2,-1). Find the coordinates of other two vertices.

Assume A(2,1)A(-2, -1) and C(3,4)C(3, 4) are two opposite vertices of a square 


AC=(3(2),4(1))\overrightarrow{AC}=(3-(-2), 4-(-1))

AC=52+52=52|\overrightarrow{AC}|=\sqrt{5^2+5^2}=5\sqrt{2}

The side of the square is 5.


slopeAC=4(1)3(2)=1slope_{AC}=\dfrac{4-(-1)}{3-(-2)}=1

Then


slopeBD=1slope_{BD}=-1

The equation of the BDBD is y=x+b.y=-x+b.

The center of the square is O(322,412).O(\dfrac{3-2}{2},\dfrac{4-1}{2} ).

Substitute


32=12+b=>b=2\dfrac{3}{2}=-\dfrac{1}{2}+b=>b=2

BD:y=x+2BD:y=-x+2


The angle between ABAB and BDBD is 45°.45\degree. Then the equation of ABAB is x=2.x=-2.


y=(2)+2=4y=-(-2)+2=4

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

The angle between CDCD and BDBD is 45°.45\degree. Then the equation of CDCD is x=3.x=3.


y=3+2=1y=-3+2=-1

D(3,1)D(3, -1)



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