Question 1

Which of the following best describes the coordinates of a point in the Cartesian plane?

Accepted Answer

A pair of real numbers $$(x, y)$$ representing the horizontal and vertical positions.

Question 2

Given points $$A = (2, 3)$$ and $$B = (6, 7)$$, what is the distance between them?

Accepted Answer

$$\sqrt{16 + 16} = \sqrt{32} \approx 5.66$$

Question 3

Which formula correctly gives the midpoint $$M of a $$segment with endpoints $$P_1$$ = ($$x_1$$, $$y_1$$)$$ and P_2$ = (x_2$, y_2$)$$?

Accepted Answer

$$M = \left( \dfrac{x_1$ + x_2$}{2}, \dfrac{y_1$ + y_2$}{2} \right)$$

Question 4

A city plans to build a hospital exactly halfway between two towns located at $$(2, 8)$$ and $$(10, 4). $$What are the coordinates of the hospital?

Accepted Answer

$$(6, 6)$$

Question 5

If the distance between points $$P = (a, b)$$ and $$Q = (a, -b) is$$ $$10$$, what is the value of $$b$$?

Accepted Answer

$$5$$

Question 6

Which of the following statements about the quadrants in the $$xy-$$plane is correct?

Accepted Answer

In Quadrant II, $$x  0.$$

Question 7

A drone flies from point $$A = (1, 2) to $$point $$B = (7, 14). $$What is the straight-line distance it travels?

Accepted Answer

$$\sqrt{(7-1)^2$ + (14-2)^2$} = \sqrt{36 + 144} = \sqrt{180} \approx 13.42$$

Question 8

Which of the following is the correct distance formula between two points $$P_1$$ = ($$x_1$$, $$y_1$$)$$ and P_2$ = (x_2$, y_2$)$$?

Accepted Answer

$$d = \sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$

Question 9

A rectangular park has its opposite corners at $$(1, 1)$$ and $$(7, 5). $$What is the length of the diagonal of the park?

Accepted Answer

$$\sqrt{(7-1)^2$ + (5-1)^2$} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21$$

Question 10

Given points $$A = (3, 4)$$, $$B = (3, 10)$$, and $$C = (9, 4)$$, what is the area of triangle $$ABC$$?

Accepted Answer

$$18$$ square units

Question 11

A cell phone tower is located at $$(0, 0). A $$user is at $$(6, 8). $$What is the distance between the user and the tower?

Accepted Answer

$$10$$

Question 12

If the midpoint of a segment is $$(5, -2)$$ and one endpoint is $$(2, 4)$$, what is the other endpoint?

Accepted Answer

$$(8, -8)$$

Question 13

A triangle has vertices at $$A = (0, 0)$$, $$B = (0, 6)$$, and $$C = (8, 0). $$What is the length of the hypotenuse $$BC$$?

Accepted Answer

$$10$$

Question 14

Which of the following best describes the abscissa of a point $$(x, y)$$?

Accepted Answer

The $$x-$$coordinate of the point

Question 15

A wildlife biologist tags two animals at $$(2, 5)$$ and $$(8, 11). $$What is the midpoint of the line segment joining their locations?

Accepted Answer

$$(5, 8)$$

Question 16

A triangle has sides of lengths $$3$$, $$4$$, and $$5. If $$the triangle is plotted in the $$xy-$$plane with vertices at $$(0, 0)$$, $$(3, 0)$$, and $$(3, 4)$$, what is the area of the triangle?

Accepted Answer

$$6$$ square units

Question 17

A delivery robot moves from $$(1, 2) to$$ $$(4, 6)$$, then to $$(7, 2). $$What is the total distance traveled by the robot?

Accepted Answer

$$5 + 5 = 10$$

Question 18

If the endpoints of a diameter of a circle are $$(-3, 4)$$ and $$(5, -2)$$, what is the center of the circle?

Accepted Answer

$$(1, 1)$$

Question 19

A rectangular field has corners at $$(0, 0)$$ and $$(a, b). $$What is the length of the diagonal of the field?

Accepted Answer

$$\sqrt{a^2$ + b^2$}$$

Distance and Midpoint Formulas in the Cartesian Plane

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