Question

question 1
a quadrilateral has vertices with coordinates (-3, 1), (0, 3), (5, 2) and (-1, -2). which type of quadrilateral is this?
rectangle
rhombus
trapezoid
square
question 2
given: in parallelogram abcd,
not drawn to scale.
what is the measurement of ∠ a?

Explanation:

Step1: Calculate side lengths (Question1)

Vertices: \(A(-3,1), B(0,3), C(5,2), D(-1,-2)\).
\(AB=\sqrt{(0+3)^2+(3-1)^2}=\sqrt{13}\), \(BC=\sqrt{(5-0)^2+(2-3)^2}=\sqrt{26}\), \(CD=\sqrt{(-1-5)^2+(-2-2)^2}=\sqrt{52}=2\sqrt{13}\), \(DA=\sqrt{(-3+1)^2+(1+2)^2}=\sqrt{13}\).

Step2: Check slopes (Question1)

Slope \(AB=\frac{3-1}{0+3}=\frac{2}{3}\), slope \(BC=\frac{2-3}{5-0}=-\frac{1}{5}\), slope \(CD=\frac{-2-2}{-1-5}=\frac{2}{3}\), slope \(DA=\frac{1+2}{-3+1}=-\frac{3}{2}\).
Parallel sides: \(AB \parallel CD\) (same slope). No other parallel sides.

Step3: Set up equations (Question2)

Opposite angles equal: \(7y+13=10y-32\); consecutive angles supplementary: \(7y+13+106-2x=180\).

Step4: Solve for y (Question2)

\(7y+13=10y-32 \Rightarrow 3y=45 \Rightarrow y=15\).

Step5: Calculate ∠A (Question2)

\(\angle A=7y+13=7(15)+13=118\).

Answer:

Question1: Trapezoid
Question2: 118