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Question

Show that the following points are the vertices of a square.

(i) A(3, 2), B(0, 5), C(−3, 2) and D(0, −1)
(ii) A(6, 2), B(2, 1), C(1, 5) and D(5, 6)
(iii) P(0, −2), Q(3, 1), R(0, 4) and S(−3, 1)

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Solution

(i) The given points are A(3, 2), B(0, 5), C(−3, 2) and D(0, −1).
AB = 0-32+5-22 = -32+32 = 9+9 = 18 =32 unitsBC = -3-02+2-52 = -32+-32 = 9+9 = 18 =32 unitsCD = 0+32+-1-22 = 32+-32 = 9+9 = 18 =32 unitsDA =0-32+-1-22 = -32+-32 = 9+9 = 18 =32 unitsTherefore, AB = BC = CD = DA = 32 unitsAlso, AC = -3-32+2-22 = -62+02 = 36 = 6 unitsBD = 0-02+-1-52 = 02+-62 = 36 = 6 unitsThus, diagonal AC = diagonal BD
Therefore, the given points form a square.


(ii) The given points are A(6, 2), B(2, 1), C(1, 5) and D(5, 6).
AB = 2-62+1-22 = -42+-12 = 16+1 = 17 unitsBC = 1-22+5-12 = -12+-42 = 1+16 = 17 unitsCD = 5-12+6-52 = 42+12 = 16+1 = 17 unitsDA =5-62+6-22 = 12+42 = 1+16 = 17 unitsTherefore AB = BC = CD = DA = 17 unitsAlso AC = 1-62+5-22 = -52+32 = 25+9 = 34 unitsBD = 5-22+6-12 = 32+52 = 9+25 = 34 unitsThus, diagonal AC = diagonal BD
Therefore, the given points form a square.


(iii) The given points are P(0, −2), Q(3, 1), R(0, 4) and S(−3, 1).
PQ = 3-02+1+22 = 32+32 = 9+9 = 18 = 32 unitsQR = 0-32+4-12 = -32+32 = 9+9 = 18 =32 unitsRS = -3-02+1-42 = -32+-32 = 9+9 = 18 = 32unitsSP =-3-02+1+22 = -32+32 = 9+9 = 18 = 32unitsTherefore, PQ = QR = RS = SP = 32 unitsAlso, PR = 0-02+4+22 = 02+62 = 36 = 6 unitsQS= -3-32+1-12 = -62+02 = 36 = 6 unitsThus, diagonal PR = diagonal QS
Therefore, the given points form a square.

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