Prove that the points [3,2], [5,4], [3,6] and [1,4] taken in order form a square. Find the area of the square.
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1
Answer:
Let A(x
1
,y
1
)=(−1,−8),B(x
2
,y
2
)=(−4,−6) and C(x
3
,y
3
)=(2,−1) and D(x
4
,y
4
)=(−3,−3) be the vertices of a square.
∴ Distance between two points =
(x
2
−x
1
)
2
+(y
2
−y
1
)
2
AB=
(4+1)
2
+(−6+8)
2
=
25+4
=
29
units
BC=
(2−4)
2
+(−1+6)
2
=
4+25
=
29
units
CD=
(−3−2)
2
+(−3+1)
2
=
25+4
=
29
units
DA=
(−3+1)
2
+(−3+8)
2
=
4+25
=
29
units
AB=CD=BC=DA
Hence, the given points form a square.
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4
There are many sports that utilize geometric shapes to help mark out the specific areas of play. Take a look at the soccer pitch below, the field of play is made up of quadrilaterals, rectangles, 90 degree angles, and circles. Furthermore, these soccer pitches, tennis courts, and basketball courts have mirror symmetry.
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