Proving Theorems with Coordinate Geometry - Geometry

1. Identify whether $(2,1)$ lies on the circle $x^2+y^2=4$.

   No, since $2^2+1^2=5\neq^4$. Substitution: $4 + 1 = 5 \neq 4$.

2. Which option is the slope of the line through $ (1,-2) $ and $ (5,6) $?

   $m=2$. $m = \frac{6-(-2)}{5-1} = \frac{8}{4} = 2$

3. What is the midpoint formula of segment endpoints $ ($x_1$,$y_1$) $ and $ ($x_2$,$y_2$) $?

   $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average of the $x$-coordinates and $y$-coordinates.

4. What slope relationship shows $\overline{AB}$ is perpendicular to $\overline{BC}$?

   $m_{AB}m_{BC}=-1$ (or one vertical, one horizontal). Product of slopes equals $-1$ for perpendicular segments.

5. What condition on slopes proves two nonvertical lines are perpendicular?

   $m_1m_2=-1$. Negative reciprocal slopes indicate perpendicular lines.

6. Identify whether $A(0,0),B(2,4),C(6,2),D(4,-2)$ form a parallelogram using midpoints.

   Yes, diagonals share midpoint $\left(3,1\right)$. Both diagonals have same midpoint.

7. Identify whether $\overline{AB}$ is perpendicular to $\overline{BC}$ for $A(0,0),B(4,0),C(4,3)$.

   Yes, horizontal is perpendicular to vertical. Horizontal and vertical segments are perpendicular.

8. What diagonal condition proves a parallelogram is a rectangle?

   Diagonals are congruent: $AC=BD$. Equal diagonal lengths characterize rectangles.

9. What is the equation of a circle centered at the origin with radius $r$?

   $x^2+y^2=r^2$. Standard form of circle equation at origin.

10. What coordinate conditions prove a quadrilateral is a square most directly?

    Rectangle and rhombus conditions both hold. Square has both rectangle and rhombus properties.

11. Identify whether $A(0,0),B(1,2),C(3,3),D(2,1)$ form a parallelogram using slopes.

    Yes, $m_{AB}=2=m_{CD}$ and $m_{BC}=\frac{1}{2}=m_{AD}$. Opposite sides have equal slopes.

12. Identify the radius of the circle centered at $(0,0)$ passing through $(0,2)$.

    $r=2$. Distance from origin to point $(0,2)$ is $2$.

13. Identify whether lines with slopes $2$ and $-\frac{1}{2}$ are perpendicular.

    Yes, they are perpendicular. Product $2 \cdot (-\frac{1}{2}) = -1$ confirms perpendicular.

14. Identify whether lines with slopes $\frac{5}{2}$ and $\frac{2}{5}$ are perpendicular.

    No, since $\frac{5}{2}\cdot\frac{2}{5}=1$. Product equals $1$, not $-1$, so not perpendicular.

15. Identify whether $(6,3)$ lies on $(x-3)^2+(y+1)^2=25$.

    Yes, since $(6-3)^2+(3+1)^2=25$. Substitution: $9 + 16 = 25$ satisfies equation.

16. Which option is the slope of the line through $(1,-2)$ and $(5,6)$?

    $m=2$. $m = \frac{6-(-2)}{5-1} = \frac{8}{4} = 2$

17. Identify whether $A(0,0),B(4,0),C(4,3),D(0,3)$ form a rectangle.

    Yes, opposite sides parallel and adjacent sides perpendicular. Rectangle has opposite sides parallel and right angles.

18. Identify whether $\overline{AB}\cong\overline{CD}$ for $A(0,0),B(3,4),C(1,1),D(4,5)$.

    Yes, both lengths are $5$. Both distances equal $5$, so segments are congruent.

19. Identify the slope of diagonal $BD$ for $B(2,2)$ and $D(2,-2)$.

    Undefined. Vertical diagonal has undefined slope.

20. Identify whether $A(0,0),B(3,1),C(4,4),D(1,3)$ is a rectangle (final classification).

    No, since $m_{AB}m_{BC}=\frac{1}{3}\cdot^3=1\neq-1$. Adjacent sides not perpendicular, so not rectangle.

21. Identify whether $A(0,0),B(2,2),C(4,0),D(2,-2)$ is a square (final classification).

    Yes, all sides equal and $m_{AB}m_{BC}=-1$. Square has equal sides and perpendicular adjacent sides.

22. Identify the slope of $\overline{BC}$ for $B(4,0)$ and $C(4,3)$.

    Undefined. Vertical segment has undefined slope.

23. Identify whether $\overline{CD}$ is vertical if $C(2,-1)$ and $D(2,9)$.

    Yes, since $x_1=x_2$. Same $x$-coordinate indicates vertical segment.

24. What diagonal condition proves a parallelogram is a rhombus?

    Diagonals are perpendicular: $m_{AC}m_{BD}=-1$. Perpendicular diagonals characterize rhombuses.

25. What two coordinate facts prove a quadrilateral is a parallelogram?

    Both pairs of opposite sides are parallel. Opposite sides parallel defines a parallelogram.

26. What slope indicates a vertical line?

    Slope is undefined; $x=\text{constant}$. No horizontal change means vertical line.

27. Identify the equation of the circle centered at $(0,0)$ passing through $(3,4)$.

    $x^2+y^2=25$. Circle through $(3,4)$ has radius $5$.

28. Identify the midpoint of $( -2,5)$ and $(4,1)$.

    $\left(1,3\right)$. $\left(\frac{-2+4}{2}, \frac{5+1}{2}\right) = (1,3)$

29. What is the equation of a circle centered at $(h,k)$ with radius $r$?

    $(x-h)^2+(y-k)^2=r^2$. Standard form of circle equation with center $(h,k)$.

30. Identify the slope of diagonal $AC$ for $A(0,0)$ and $C(4,0)$.

    $m_{AC}=0$. Horizontal diagonal has slope $0$.

31. Identify the equation of the circle centered at $ (0,0) $ through $ (0,2) $.

    $x^2+y^2=4$. Circle centered at origin with radius $2$.

32. Identify the midpoint of diagonal endpoints $A(0,0)$ and $C(6,2)$.

    $\left(3,1\right)$. Midpoint of diagonal $AC$.

33. What coordinate condition proves $\overline{AB}$ bisects $\overline{CD}$?

    Midpoint of $CD$ lies on line $AB$. Bisector passes through the midpoint of the segment.

34. Identify the slope of the line through $(2,3)$ and $(6,11)$.

    $m=2$. $m = \frac{11-3}{6-2} = \frac{8}{4} = 2$

35. What slope indicates a horizontal line?

    $m=0$. No vertical change means horizontal line.

36. Identify whether diagonals $AC$ and $BD$ are perpendicular if one is horizontal and one is vertical.

    Yes, they are perpendicular. Horizontal and vertical lines are perpendicular.

37. Identify whether $(4,4)$ lies on the circle centered at origin through $(3,4)$.

    No, since $4^2+4^2=32\neq25$. Point $(4,4)$ does not satisfy equation.

38. Identify whether $A(0,0),B(2,1),C(4,0),D(2,-1)$ form a rectangle using slopes.

    No, adjacent slopes are $\frac{1}{2}$ and $-\frac{1}{2}$ (not perpendicular). Adjacent slopes not negative reciprocals.

39. Identify whether $(5,0)$ lies on the circle $x^2+y^2=25$.

    Yes, since $5^2+0^2=25$. Point $(5,0)$ satisfies the circle equation.

40. What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Pythagorean theorem applied to coordinate differences.

41. Identify whether $(1,\sqrt{3})$ lies on $x^2+y^2=4$.

    Yes, since $1^2+(\sqrt{3})^2=4$. Substitution: $1 + 3 = 4$ satisfies equation.

42. Identify the distance between $(0,0)$ and $(3,4)$.

    $5$. $d = \sqrt{3^2+4^2} = \sqrt{25} = 5$

43. Identify the equation of the circle centered at $(0,0)$ through $(0,2)$.

    $x^2+y^2=4$. Circle centered at origin with radius $2$.

44. Identify the equation of the circle centered at $(3,-1)$ with radius $5$.

    $(x-3)^2+(y+1)^2=25$. Standard form with center $(3,-1)$ and radius $5$.

45. Identify the midpoint of $(-2,5)$ and $(4,1)$.

    $\left(1,3\right)$. $\left(\frac{-2+4}{2}, \frac{5+1}{2}\right) = \left(1,3\right)$

46. Identify the diagonal lengths for rectangle $A(0,0),B(4,0),C(4,3),D(0,3)$.

    $AC=BD=5$. Both diagonals have length $5$.

47. Identify the midpoint of diagonal endpoints $B(2,4)$ and $D(4,-2)$.

    $\left(3,1\right)$. Midpoint of diagonal $BD$.

48. What algebraic test determines whether $(x,y)$ lies on circle $x^2+y^2=r^2$?

    Check whether $x^2+y^2=r^2$. Substitute coordinates into circle equation.

49. What coordinate fact proves a quadrilateral is a parallelogram using diagonals?

    Its diagonals have the same midpoint. Bisecting diagonals characterize parallelograms.

50. Identify whether $A(-1,0),B(1,0),C(1,2),D(-1,2)$ is a rectangle.

    Yes, sides are horizontal/vertical with right angles. All sides horizontal or vertical with right angles.

51. Identify $AB$ for $A(1,2)$ and $B(4,6)$.

    $5$. $d = \sqrt{(4-1)^2+(6-2)^2} = \sqrt{25} = 5$

52. What condition on slopes proves two nonvertical lines are parallel?

    $m_1=m_2$. Equal slopes indicate parallel lines.

53. Identify the slope of $\overline{AB}$ for $A(0,0)$ and $B(4,0)$.

    $m_{AB}=0$. Horizontal segment has slope $0$.

54. What coordinate conditions prove a quadrilateral is a rhombus?

    All four sides congruent: $AB=BC=CD=DA$. All sides equal defines a rhombus.

55. What coordinate conditions prove a quadrilateral is a rectangle using slopes?

    Opposite sides parallel and one right angle. Parallelogram with one right angle is a rectangle.

56. Identify whether $A(0,0),B(2,2),C(4,0),D(2,-2)$ form a rhombus using side lengths.

    Yes, all sides have length $\sqrt{8}$. All four side lengths are equal.

57. Identify whether the parallelogram $A(0,0),B(1,2),C(3,3),D(2,1)$ is a rectangle.

    No, since $m_{AB}m_{BC}=1\neq-1$. Adjacent slopes multiply to $1$, not $-1$.

58. Identify whether lines with slopes $\frac{3}{4}$ and $\frac{3}{4}$ are parallel.

    Yes, they are parallel. Equal slopes confirm parallel lines.

59. Identify $BD$ for $B(0,3)$ and $D(4,3)$.

    $4$. Distance between points $B$ and $D$.

60. What is the midpoint formula of segment endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

    $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average of the $x$-coordinates and $y$-coordinates.

61. What coordinate condition proves diagonals of a quadrilateral bisect each other?

    Diagonals have the same midpoint. Shared midpoint means diagonals bisect each other.

62. What is the slope formula for points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.

63. What coordinate condition proves two segments $\overline{AB}$ and $\overline{CD}$ are congruent?

    $AB=CD$ using the distance formula. Equal distances prove segment congruence.

64. Identify whether $\overline{AB}$ is horizontal if $A(-3,7)$ and $B(5,7)$.

    Yes, since $y_1=y_2$. Same $y$-coordinate indicates horizontal segment.

65. Identify $AC$ for $A(0,0)$ and $C(4,0)$.

    $4$. Distance between points $A$ and $C$.

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