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Mathematics · Coordinate Geometry · NCERT Examples

Example 2

A complete, independently verified solution for NCERT Example 2.

Mathematics · Chapter 7 · NCERT Examples

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NCERT Example

Example 2

distance formula · square · diagonals

Show that the points A(1, 7), B(4, 2), C(-1, -1) and D(-4, 4) are the vertices of a square.

Solution

Board-exam working:

Using the distance formula for each side:

AB=(4−1)2+(2−7)2=9+25=34AB=\sqrt{(4-1)^2+(2-7)^2}=\sqrt{9+25}=\sqrt{34}
BC=(−1−4)2+(−1−2)2=25+9=34BC=\sqrt{(-1-4)^2+(-1-2)^2}=\sqrt{25+9}=\sqrt{34}
CD=(−4−(−1))2+(4−(−1))2=9+25=34CD=\sqrt{(-4-(-1))^2+(4-(-1))^2}=\sqrt{9+25}=\sqrt{34}
DA=(1−(−4))2+(7−4)2=25+9=34DA=\sqrt{(1-(-4))^2+(7-4)^2}=\sqrt{25+9}=\sqrt{34}

Now calculate the diagonals:

AC=(−1−1)2+(−1−7)2=4+64=68AC=\sqrt{(-1-1)^2+(-1-7)^2}=\sqrt{4+64}=\sqrt{68}
BD=(−4−4)2+(4−2)2=64+4=68BD=\sqrt{(-4-4)^2+(4-2)^2}=\sqrt{64+4}=\sqrt{68}
AB=BC=CD=DAAB=BC=CD=DA
AC=BDAC=BD

All four sides are equal and both diagonals are equal. Therefore, ABCD is a square.

Final answer

ABCD is a square.