To determine the shape formed by the points [tex]\((-4, 8)\)[/tex], [tex]\((9, 16)\)[/tex], [tex]\((3, -17)\)[/tex], and [tex]\((-9, -6)\)[/tex], we need to analyze the distances between them. Here is a step-by-step approach:
1. Plotting the Points on a Coordinate Grid:
- First, we plot the points: [tex]\((-4, 8)\)[/tex], [tex]\((9, 16)\)[/tex], [tex]\((3, -17)\)[/tex], and [tex]\((-9, -6)\)[/tex].
2. Label the Points:
- Let's label the points as follows:
- [tex]\(A = (-4, 8)\)[/tex]
- [tex]\(B = (9, 16)\)[/tex]
- [tex]\(C = (3, -17)\)[/tex]
- [tex]\(D = (-9, -6)\)[/tex]
3. Calculate the Distances Between Points:
- We need to calculate the distances between consecutive points and the diagonals to understand the shape.
- Distance between [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
AB = \sqrt{(9 - (-4))^2 + (16 - 8)^2} = \sqrt{13^2 + 8^2} = \sqrt{169 + 64} = \sqrt{233} \approx 15.26
\][/tex]
- Distance between [tex]\(B\)[/tex] and [tex]\(C\)[/tex]:
[tex]\[
BC = \sqrt{(3 - 9)^2 + (-17 - 16)^2} = \sqrt{(-6)^2 + (-33)^2} = \sqrt{36 + 1089} = \sqrt{1125} \approx 33.54
\][/tex]
- Distance between [tex]\(C\)[/tex] and [tex]\(D\)[/tex]:
[tex]\[
CD = \sqrt{(3 - (-9))^2 + (-17 - (-6))^2} = \sqrt{12^2 + (-11)^2} = \sqrt{144 + 121} = \sqrt{265} \approx 16.28
\][/tex]
- Distance between [tex]\(D\)[/tex] and [tex]\(A\)[/tex]:
[tex]\[
DA = \sqrt{(-9 - (-4))^2 + (-6 - 8)^2} = \sqrt{(-5)^2 + (-14)^2} = \sqrt{25 + 196} = \sqrt{221} \approx 14.87
\][/tex]
- Diagonal [tex]\(AC\)[/tex]:
[tex]\[
AC = \sqrt{(3 - (-4))^2 + (-17 - 8)^2} = \sqrt{7^2 + (-25)^2} = \sqrt{49 + 625} = \sqrt{674} \approx 25.96
\][/tex]
- Diagonal [tex]\(BD\)[/tex]:
[tex]\[
BD = \sqrt{(9 - (-9))^2 + (16 - (-6))^2} = \sqrt{18^2 + 22^2} = \sqrt{324 + 484} = \sqrt{808} \approx 28.43
\][/tex]
4. Summarize the Distances:
- The distances calculated are approximately:
- [tex]\(AB \approx 15.26\)[/tex]
- [tex]\(BC \approx 33.54\)[/tex]
- [tex]\(CD \approx 16.28\)[/tex]
- [tex]\(DA \approx 14.87\)[/tex]
- Diagonal [tex]\(AC \approx 25.96\)[/tex]
- Diagonal [tex]\(BD \approx 28.43\)[/tex]
5. Determine the Shape:
- For the points to form a regular shape like a rectangle or square, the opposite sides and the diagonals should have specific equalities.
- In this case:
- [tex]\(AB\)[/tex] and [tex]\(CD\)[/tex] are not equal.
- [tex]\(BC\)[/tex] and [tex]\(DA\)[/tex] are not equal.
- The diagonals [tex]\(AC\)[/tex] and [tex]\(BD\)[/tex] are not equal.
Given these observations, the distances do not correspond to any regular shape.
Conclusion:
The points [tex]\((-4, 8)\)[/tex], [tex]\((9, 16)\)[/tex], [tex]\((3, -17)\)[/tex], and [tex]\((-9, -6)\)[/tex] form an Irregular Quadrilateral.
