Show that (9, 0)(9, 6)(-9, 6)and (-9, 0)are the vertices of a rectangle.
Answers
Answer:
Solution
Given
the points are (9,0) , (9,6) , (-9,6) and (-9,0)
Let us consider these points as A , B ,C and D respectively .
So that ,
• A(9,0)
• B(9,6)
• C(-9,6)
• D(-9,0)
Formula to be used
\begin{gathered}Distance \: between \: two\: points \: ( x_{1},y_{1}) \: and \: ( x_{2},y_{2})is \\ = \sqrt{ (x_{2} - x_{1}) ^{2} + (y_{2} - y_{1})^{2} } \end{gathered}
Distancebetweentwopoints(x
1
,y
1
)and(x
2
,y
2
)is
=
(x
2
−x
1
)
2
+(y
2
−y
1
)
2
So Applying the distance formula in the given Data we have
\begin{gathered}AB = \sqrt{(9 - 9) {}^{2} + (6 - 0) {}^{2} } \\ \implies AB = \sqrt{36} \\ \implies AB = 6 \: units\end{gathered}
AB=
(9−9)
2
+(6−0)
2
⟹AB=
36
⟹AB=6units
\begin{gathered}BC = \sqrt{( - 9 - 9) ^{2} + (6 - 6) {}^{2} } \\ \implies BC = \sqrt{( - 18) ^{2} } \\ \implies BC = \sqrt{( - 1) ^{2} \times ( {18})^{2} } \\ \implies BC = 18 \: units\end{gathered}
BC=
(−9−9)
2
+(6−6)
2
⟹BC=
(−18)
2
⟹BC=
(−1)
2
×(18)
2
⟹BC=18units
\begin{gathered}CD = \sqrt{( - 9 + 9) {}^{2} + (0 - 6) ^{2} } \\ \implies CD = \sqrt{( - 6) ^{2} } \\ \implies CD = \sqrt{ {( - 1)}^{2} \times (6) {}^{2} } \\ \implies CD = 6 \: units\end{gathered}
CD=
(−9+9)
2
+(0−6)
2
⟹CD=
(−6)
2
⟹CD=
(−1)
2
×(6)
2
⟹CD=6units
\begin{gathered}AD = \sqrt{( - 9 - 9)^{2} + (0 - 0) ^{2} } \\ \implies AD = \sqrt{( - 18) {}^{2} } \\ \implies AD = \sqrt{( { - 1)}^{2} \times (18) {}^{2} } \\ \implies AD = 18 \: units\end{gathered}
AD=
(−9−9)
2
+(0−0)
2
⟹AD=
(−18)
2
⟹AD=
(−1)
2
×(18)
2
⟹AD=18units
Thus the opposite sides
AB = CD = 6 units
BC = AD = 18 units
Now for diagonals
\begin{gathered}AC = \sqrt{( - 9 - 9) ^{2} + (6 - 0) ^{2} } \\ \implies AC = \sqrt{(18 )^{2} + (6 )^{2} } \\ \implies AC = \sqrt{324 + 36} \\ \implies AC = \sqrt{360} \\ \implies AC = \sqrt{36 \times 10} \\ \implies AC = 6 \sqrt{10} \: units\end{gathered}
AC=
(−9−9)
2
+(6−0)
2
⟹AC=
(18)
2
+(6)
2
⟹AC=
324+36
⟹AC=
360
⟹AC=
36×10
⟹AC=6
10
units
\begin{gathered}BD = \sqrt{(9 + 9) ^{2} + (0 - 6) ^{2} } \\ \implies BD = \sqrt{(18 {)}^{2} + ( {6})^{2} } \\ \implies BD = \sqrt{324 + 36} \\ \implies BD = \sqrt{360} \\ \implies BD = 6 \sqrt{10} \: units\end{gathered}
BD=
(9+9)
2
+(0−6)
2
⟹BD=
(18)
2
+(6)
2
⟹BD=
324+36
⟹BD=
360
⟹BD=6
10
units
Thus the diagonals AC = BD = 6√10 units
Now from the properties of a rectangle we have
• The opposite sides of a rectangle are equal
• The diagonals of a rectangle are equal
Therefore , ABCD is a rectangle
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