Coordinate Geometry and Distance - Quantitative Reasoning

1. What is the distance between $(-1,-2)$ and $(2,2)$?

   $5$. Distance formula results in $sqrt{(2-(-1))^2+(2-(-2))^2}=sqrt{9+16}=5$.

2. What is the midpoint of the segment joining $(-3,5)$ and $(1,-1)$?

   $\left(-1,2\right)$. Midpoint formula: left(\frac{-3+1}{2},\frac{5+(-1)}{2}\right)=(-1,2).

3. What is the distance between $(1,2)$ and $(4,6)$?

   $5$. Distance formula gives $sqrt{(4-1)^2+(6-2)^2}=sqrt{9+16}=5$.

4. State the formula for the midpoint of the segment joining $(x_1,y_1)$ and $(x_2,y_2)$.

   $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Averages the x-coordinates and y-coordinates separately to find the central point of the line segment.

5. What is the equation of the circle with center $(0,0)$ and radius $r$?

   $x^2+y^2=r^2$. Special case of the circle equation when the center is at the origin, using distance from $(0,0)$.

6. What is the distance from $(4,-3)$ to the origin $(0,0)$?

   $5$. Distance to origin: $sqrt{(4-0)^2+(-3-0)^2}=sqrt{16+9}=5$.

7. Identify the condition for points $(x_1,y_1)$ and $(x_2,y_2)$ to be exactly $r$ units apart.

   $(x_2-x_1)^2+(y_2-y_1)^2=r^2$. Sets the squared distance equal to $r^2$ to define points exactly $r$ units apart without needing the square root.

8. What is the distance between $(5,-1)$ and $(5,6)$?

   $7$. Vertical line segment with $Delta x=0$ and $Delta y=7$, so distance equals $|6-(-1)|$.

9. What is the distance between $(2,3)$ and $(8,3)$?

   $6$. Horizontal line segment with $Delta x=6$ and $Delta y=0$, so distance equals $|8-2|$.

10. What is the equation of the circle with center $(h,k)$ and radius $r$?

    $(x-h)^2+(y-k)^2=r^2$. Represents all points $(x,y)$ at distance $r$ from center $(h,k)$ using the distance formula squared.

11. State the formula for the distance from the origin $(0,0)$ to a point $(x,y)$.

    $\sqrt{x^2+y^2}$. Applies the distance formula with $(x_1,y_1)=(0,0)$, yielding the magnitude of the position vector from the origin.

12. What is the distance from $(x_1,y_1)$ to $(x_1,y_2)$ (same $x$-coordinate)?

    $|y_2-y_1|$. With identical x-coordinates, the distance simplifies to the absolute difference in y-coordinates, as the horizontal change is zero.

13. What is the distance from point $(x_0,y_0)$ to the horizontal line $y=c$?

    $|y_0-c|$. Vertical distance between the point's y-coordinate and the line $y=c$ uses the absolute difference.

14. State the distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane.

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Derived from the Pythagorean theorem, calculating the hypotenuse of the right triangle formed by the horizontal and vertical differences in coordinates.

15. What is the midpoint of the segment joining $(2,7)$ and $(6,1)$?

    $\left(4,4\right)$. Midpoint formula: left(\frac{2+6}{2},\frac{7+1}{2}\right)=(4,4).

16. What is the squared distance between $(x_1,y_1)$ and $(x_2,y_2)$ (no square root)?

    $(x_2-x_1)^2+(y_2-y_1)^2$. Squares the differences in coordinates to compute distance squared, avoiding the square root for comparisons or equations.

17. What is the distance between $(-2,1)$ and $(1,5)$?

    $5$. Distance formula yields $sqrt{(1-(-2))^2+(5-1)^2}=sqrt{9+16}=5$.

18. What is the distance from $(x_1,y_1)$ to $(x_2,y_1)$ (same $y$-coordinate)?

    $|x_2-x_1|$. With identical y-coordinates, the distance simplifies to the absolute difference in x-coordinates, as the vertical change is zero.

19. What is the distance from $(x,y)$ to the $x$-axis?

    $|y|$. Vertical distance from any point on the line $y=y$ to $y=0$ is the absolute value of the y-coordinate.

20. What is the length of the diagonal of a rectangle with corners $(0,0)$ and $(6,8)$?

    $10$. Diagonal length is the distance between opposite corners: $sqrt{(6-0)^2+(8-0)^2}=sqrt{36+64}=10$.

21. What is the distance from $(x,y)$ to the $y$-axis?

    $|x|$. Horizontal distance from any point on the line $x=x$ to $x=0$ is the absolute value of the x-coordinate.

22. What is the distance between $(0,0)$ and $(3,4)$?

    $5$. Applies the distance formula: $sqrt{(3-0)^2+(4-0)^2}=sqrt{9+16}=sqrt{25}=5$.

23. What is the distance from point $(x_0,y_0)$ to the vertical line $x=c$?

    $|x_0-c|$. Horizontal distance between the point's x-coordinate and the line $x=c$ uses the absolute difference.

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