Question

question 5
what is the distance formula?
a. $sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
b. $sqrt{(x_2-x_1)+(y_2-y_1)}$
c. $sqrt{(y_2-y_1)^2-(x_2-x_1)^2}$
d. $sqrt{(x_2+x_1)^2-(y_2+y_1)^2}$

question 6
is the following shape a rectangle? how do you know?
a. no, because the figure is not a quadrilateral.
b. there is not enough information to determine.
c. yes, the adjacent sides are perpendicular, and the opposite sides are parallel.
d. no, the adjacent sides are not perpendicular.

question 7
is the following shape a right triangle? how do you know?
a. no, there is no right angle.
b. there is not enough information to determine.
c. no, the side lengths do not fit the pythagorean theorem.
d. yes, two sides are perpendicular, and the side lengths fit the pythagorean theorem.

Explanation:

Step1: Identify distance formula

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ comes from the Pythagorean theorem, squaring the differences in coordinates, summing them, then taking the square root.

Step2: Analyze Question 6 shape

The shape has 4 vertices (quadrilateral). Check perpendicularity: adjacent sides do not form right angles (slopes don't multiply to -1, or angles are not 90°).

Step3: Analyze Question 7 triangle

Check if sides fit Pythagoras, or if two sides are perpendicular. The triangle has a pair of perpendicular sides, and side lengths satisfy $a^2+b^2=c^2$.

Answer:

Question 5: A. $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Question 6: D. No, the adjacent sides are not perpendicular.
Question 7: D. Yes, two sides are perpendicular, and the side lengths fit the Pythagorean theorem.