$$\frac{1}{2} \times |1 - 4| \times |6 - 2| = \frac{1}{2} \times 3 \times 4 = 6$$

Question 1

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

Accepted Answer

$$x \u003c 0, y \u003c 0$$

Question 2

What is the distance between the points $$A(2, 3)$$ and $$B(-1, 7) in $$the Cartesian plane?

Accepted Answer

$$\sqrt{(2 + 1)^2$ + (3 - 7)^2$} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Question 3

If a point has coordinates $$(0, 5)$$, where is it located in the Cartesian plane?

Accepted Answer

On the y-axis

Question 4

What is the midpoint of the segment joining the points $$P(4, -2)$$ and $$Q(-6, 8)$$?

Accepted Answer

$$\left(\frac{4 + (-6)}{2}, \frac{-2 + 8}{2}\right) = (-1, 3)$$

Question 5

Which statement about the distance formula is correct?

Accepted Answer

The distance between two points is always positive unless the points are identical.

Question 6

A triangle has vertices at $$A(1, 2)$$, $$B(4, 6)$$, and $$C(1, 6). $$What is the area of the triangle?

Accepted Answer

$$\frac{1}{2} 	imes |1 - 4| 	imes |6 - 2| = \frac{1}{2} 	imes 3 	imes 4 = 6$$

Question 7

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

Accepted Answer

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

Question 8

A patient’s heart monitor records two points: $$A(-3, 2)$$ and $$B(1, -4). $$What is the distance between these points?

Accepted Answer

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

Question 9

If the midpoint of a segment is $$(0, 0)$$ and one endpoint is $$(4, 6)$$, what are the coordinates of the other endpoint?

Accepted Answer

$$(-4, -6)$$

Question 10

Which of the following statements about the midpoint formula is true?

Accepted Answer

The midpoint is found by averaging the x-coordinates and y-coordinates of the endpoints.

Question 11

A chemical reaction is monitored at two locations: $$(2, 5)$$ and $$(8, 11). $$What is the midpoint between these two locations?

Accepted Answer

$$\left(\frac{2 + 8}{2}, \frac{5 + 11}{2}\right) = (5, 8)$$

Question 12

Which of the following points is located in Quadrant II?

Accepted Answer

$$(-5, 7)$$

Question 13

A city is mapped with coordinates $$(x, y). If a $$location is at $$(0, -3)$$, where is it?

Accepted Answer

On the y-axis

Question 14

Which of the following is NOT a property of the distance formula?

Accepted Answer

The distance between two points is negative if the points are in different quadrants.

Question 15

A triangle has vertices at $$A(0, 0)$$, $$B(3, 0)$$, and $$C(0, 4). $$What is the length of the hypotenuse of the triangle?

Accepted Answer

$$\sqrt{(3 - 0)^2$ + (0 - 4)^2$} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Question 16

If a point is located to the left of the origin, which statement is true?

Accepted Answer

Its x-coordinate is negative.

Question 17

A medical imaging device records two points: $$A(7, -2)$$ and $$B(-1, 4). $$What is the midpoint between these points?

Accepted Answer

$$\left(\frac{7 + (-1)}{2}, \frac{-2 + 4}{2}\right) = (3, 1)$$

Question 18

Which of the following points is located in Quadrant IV?

Accepted Answer

$$(6, -3)$$

Question 19

If the coordinates of the origin are $$(0, 0)$$, which statement is true?

Accepted Answer

Every point on the x-axis has coordinates $$(x, 0).$$

Chapter 1: Graphs – The Distance and Midpoint Formulas

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