To determine the type of quadrilateral ABCD, we will go through the properties required to classify it. The vertices of quadrilateral ABCD are given by the coordinates [tex]\( A (3,-5) \)[/tex], [tex]\( B (5,-2) \)[/tex], [tex]\( C (10,-4) \)[/tex], and [tex]\( D (8,-7) \)[/tex]. Let's analyze the quadrilateral step-by-step.
1. Calculate the lengths of the sides:
- Side [tex]\( AB \)[/tex]:
[tex]\[
AB = \sqrt{(5 - 3)^2 + (-2 - (-5))^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\][/tex]
- Side [tex]\( BC \)[/tex]:
[tex]\[
BC = \sqrt{(10 - 5)^2 + (-4 - (-2))^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}
\][/tex]
- Side [tex]\( CD \)[/tex]:
[tex]\[
CD = \sqrt{(10 - 8)^2 + (-4 - (-7))^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\][/tex]
- Side [tex]\( DA \)[/tex]:
[tex]\[
DA = \sqrt{(8 - 3)^2 + (-7 - (-5))^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}
\][/tex]
2. Check for congruence of sides:
- Opposite sides [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] are congruent ([tex]\( \sqrt{13} \)[/tex]).
- Opposite sides [tex]\( BC \)[/tex] and [tex]\( DA \)[/tex] are congruent ([tex]\( \sqrt{29} \)[/tex]).
3. Check for perpendicularity of adjacent sides:
- Vectors of the sides:
- Vector [tex]\( \overrightarrow{AB} \)[/tex]:
[tex]\[
\overrightarrow{AB} = (5 - 3, -2 - (-5)) = (2, 3)
\][/tex]
- Vector [tex]\( \overrightarrow{BC} \)[/tex]:
[tex]\[
\overrightarrow{BC} = (10 - 5, -4 - (-2)) = (5, -2)
\][/tex]
- Vector [tex]\( \overrightarrow{CD} \)[/tex]:
[tex]\[
\overrightarrow{CD} = (10 - 8, -4 - (-7)) = (2, 3)
\][/tex]
- Vector [tex]\( \overrightarrow{DA} \)[/tex]:
[tex]\[
\overrightarrow{DA} = (3 - 8, -5 - (-7)) = (-5, 2)
\][/tex]
- Check Dot Products for Perpendicularity:
- Dot product [tex]\( \overrightarrow{AB} \cdot \overrightarrow{BC} \)[/tex]:
[tex]\[
\overrightarrow{AB} \cdot \overrightarrow{BC} = 2 \cdot 5 + 3 \cdot (-2) = 10 - 6 = 4 \neq 0
\][/tex]
- Because the dot product is not zero, adjacent sides [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] are not perpendicular.
From our findings:
- Opposite sides [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] are congruent, and opposite sides [tex]\( BC \)[/tex] and [tex]\( DA \)[/tex] are congruent.
- Adjacent sides are not perpendicular.
Given these properties, Quadrilateral [tex]\(ABCD\)[/tex] is indeed a parallelogram because opposite sides are congruent, but the adjacent sides are not perpendicular.
Therefore, the correct classification for quadrilateral [tex]\(ABCD\)[/tex] is:
[tex]\[
\boxed{\text{parallelogram}}
\][/tex]
