# Show that the points a(2, 2), b(5, 7), c(-5, 13), and d(-8, 8) are the vertices of a rectangle.

**Solution:**

The points a(2, 2), b(5, 7), c(-5, 13), and d(-8, 8) are the vertices of a rectangle.

Let us prove that the opposite sides are equal.

Consider a(2, 2), b(5, 7)

x\(_1\) = 2, y\(_1\) = 2, x\(_2\)= 5, y\(_2\) = 7

AB^{2} = (x\(_2\) - x\(_1\))^{2} + (y\(_2\) - y\(_1\))^{2 }(using the distance formula between 2 points)

Substituting the values

AB^{2} = (5 - 2)^{2} + (7 - 2)^{2}

AB^{2} = (3)^{2} + (5)^{2}

By further calculation

AB^{2} = (9 + 25)

AB^{2} = 34

AB = √34 units

Consider b(5, 7), c(-5, 13)

x\(_1\) = 5, y\(_1\)= 7, x_{2} = -5, y_{2} = 13

BC^{2} = (x\(_2\)_{ }- x\(_1\))^{2} + (y\(_2\)_{ }- y\(_1\))^{2}

Substituting the values

BC^{2} = (-5 - 5)^{2} + (13 - 7)^{2}

BC^{2} = (-10)^{2} + (6)^{2}

By further calculation

BC^{2} = 100 +36

BC^{2} = 136

BC = √136 units

Consider c(-5, 13), d(-8, 8)

x_{1} = -5, y_{1} = 13, x_{2} = -8, y_{2} = 8

CD^{2} = (x\(_2\) - x\(_1\))^{2} + (y\(_2\) - y\(_1\))^{2}

Substituting the values

CD^{2} = (-8 + 5)^{2} + (8 - 13)^{2}

CD^{2} = (-3)^{2} + (-5)^{2}

By further calculation

CD^{2} = 9 + 25

CD^{2} = 34

CD = √34 units

Consider a(2, 2), d(-8, 8)

x_{1} = 2, y_{1} = 2, x_{2} = -8, y_{2} = 8

AD^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Substituting the values

AD^{2} = (-8 - 2)^{2} + (8 - 2)^{2}

AD^{2} = (-10)^{2} + (6)^{2}

By further calculation

AD^{2} = 100 +36

AD^{2} = 136

AD = √136 units

Here AB = CD = √34 units and BC = AD = √136 units

As the opposite sides are equal, ABCD is a rectangle.

Therefore, the points a(2, 2), b(5, 7), c(-5, 13), and d(-8, 8) are the vertices of a rectangle.

## Show that the points a(2, 2), b(5, 7), c(-5, 13), and d(-8, 8) are the vertices of a rectangle.

**Summary:**

The points a(2, 2), b(5, 7), c(-5, 13), and d(-8, 8) are the vertices of a rectangle

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