Question 1

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

Accepted Answer

A pair of real numbers $$(x, y)$$ that locate a point relative to the x-axis and y-axis.

Question 2

Given the points $$A = (2, -1)$$ and $$B = (-4, 3)$$, what is the distance between them?

Accepted Answer

$$\sqrt{(-4-2)^2$ + (3+1)^2$} = \sqrt{36 + 16} = \sqrt{52}$$

Question 3

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

Accepted Answer

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

Question 4

A city park is represented on a map by the points $$(1, 2)$$ and $$(7, 8). $$What are the coordinates of the park's center?

Accepted Answer

$$(4, 5)$$

Question 5

Which quadrant does the point $$(-5, 4)$$ lie in?

Accepted Answer

Quadrant II

Question 6

If the endpoints of a diameter of a circular pond are $$(2, 3)$$ and $$(8, 11)$$, what is the length of the diameter?

Accepted Answer

$$\sqrt{(8-2)^2$ + (11-3)^2$} = \sqrt{36 + 64} = 10$$

Question 7

A triangle has vertices at $$A = (0, 0)$$, $$B = (6, 0)$$, and $$C = (0, 8). Is $$this triangle a right triangle?

Accepted Answer

Yes, because $$6^2$$ + $$8^2$$ = $$10^2 by $$the Pythagorean Theorem.

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 drone flies from $$(1, 2) to$$ $$(4, 6). $$What is the straight-line distance it travels?

Accepted Answer

$$\sqrt{(4-1)^2$ + (6-2)^2$} = \sqrt{9 + 16} = 5$$

Question 10

If the midpoint of a segment is $$(5, -2)$$ and one endpoint is $$(3, 4)$$, what is the other endpoint?

Accepted Answer

$$(7, -8)$$

Question 11

Which of the following statements is true about the coordinate axes?

Accepted Answer

They divide the plane into four quadrants.

Question 12

A rectangular field has corners at $$(2, 3)$$ and $$(10, 7). $$What is the length of the diagonal?

Accepted Answer

$$\sqrt{(10-2)^2$ + (7-3)^2$} = \sqrt{64 + 16} = \sqrt{80}$$

Question 13

Which of the following points lies in Quadrant IV?

Accepted Answer

$$(5, -3)$$

Question 14

A wildlife biologist marks two locations: $$(0, 0)$$ and $$(12, 5). $$What is the midpoint between these locations?

Accepted Answer

$$(6, 2.5)$$

Question 15

If a segment has endpoints $$(-2, 7)$$ and $$(4, -1)$$, what is the midpoint?

Accepted Answer

$$(1, 3)$$

Question 16

A triangle has vertices at $$A = (1, 2)$$, $$B = (5, 2)$$, and $$C = (1, 6). $$Which side is the longest?

Accepted Answer

$$BC$$, because $$\sqrt{(5-1)^2$ + (2-6)^2$} = \sqrt{16 + 16} = \sqrt{32}$$

Question 17

Which of the following is NOT a property of the Cartesian coordinate system?

Accepted Answer

The axes are parallel to each other.

Question 18

A segment has endpoints $$(a, b)$$ and $$(c, d). $$Which formula gives the midpoint?

Accepted Answer

$$\left( \dfrac{a+c}{2}, \dfrac{b+d}{2} \right)$$

Question 19

A surveyor needs to find the distance between two points on a map: $$(7, -2)$$ and $$(-1, 3). $$What is the distance?

Accepted Answer

$$\sqrt{(7+1)^2$ + (-2-3)^2$} = \sqrt{64 + 25} = \sqrt{89}$$

Foundations: The Distance and Midpoint Formulas in the Coordinate Plane

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