The following set of coordinates represents which figure? (1, 1), (5, 3), (7, 7), (3, 5)

Solution:

Given, the points are A(1, 1) B(5, 3) C(7, 7) D(3, 5)

We have to find the figure represented by the set of coordinates.

By distance formula,

\(distance=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\)

AB = \(=\sqrt{(5-1)^{2}+(3-1)^{2}}=\sqrt{(4)^{2}+(2)^{2}}=\sqrt{16+4}=\sqrt{20}\)

BC = \(=\sqrt{(7-5)^{2}+(7-3)^{2}}=\sqrt{(2)^{2}+(4)^{2}}=\sqrt{4+16}=\sqrt{20}\)

By slope formula,

\(m=\frac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\)

Slope of AB = \(m=\frac{3-1}{5-1}=\frac{2}{4}=\frac{1}{2}\)

Slope of BC = \(m=\frac{7-3}{7-5}=\frac{4}{2}=2\)

Slope of CD = \(m=\frac{5-7}{3-7}=\frac{-2}{-4}=\frac{1}{2}\)

Slope of DA = \(m=\frac{5-1}{3-1}=\frac{4}{2}=2\)

We have 2 pairs of opposite sides that are parallel.

So, the figure can be a parallelogram or a rhombus.

From distance calculation, it is clear that adjacent sides are equal.

This implies the figure is a rhombus.

Therefore, the set of coordinates represents a rhombus.

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The following set of coordinates represents which figure? (1, 1), (5, 3), (7, 7), (3, 5)

Summary:

The following set of coordinates (1, 1), (5, 3), (7, 7), (3, 5) represent a rhombus.