Coordinates to Find Perimeter and Area - Geometry

1. Find the slope of the line through $(1,2)$ and $(5,10)$.

   $2$. $\frac{10-2}{5-1}=\frac{8}{4}=2$

2. What is the area of an axis-aligned rectangle with corners $(x_1,y_1)$ and $(x_2,y_2)$?

   $A=|x_2-x_1|\,|y_2-y_1|$. Width times height for axis-aligned rectangles.

3. What is the distance between $(x,y)$ and $(x+a,y+b)$?

   $\sqrt{a^2+b^2}$. Translation doesn't change the distance between points.

4. Find the area of the axis-aligned rectangle with corners $(0,0)$ and $(6,8)$.

   $48$. Width $6$, height $8$: area is $6 \times 8=48$.

5. What is the perimeter of a polygon in the coordinate plane in terms of side lengths?

   Sum of all side lengths (use $d$ for each side). Apply distance formula to each consecutive pair of vertices.

6. Find the area of the triangle with vertices $(0,0)$, $(4,0)$, $(4,3)$.

   $6$. Right triangle: $\frac{1}{2} \times 4 \times 3 = 6$

7. Find the distance between $(2,1)$ and $(6,1)$.

   $4$. Horizontal distance: $|6-2|=4$

8. What is the area formula for a triangle with base $b$ and height $h$?

   $A=\frac{1}{2}bh$. Triangle area is half the base times height.

9. Find the distance between $(-1,-2)$ and $(2,2)$.

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

10. What is the distance between $(0,0)$ and $(a,b)$?

    $\sqrt{a^2+b^2}$. Direct application of distance formula from origin.

11. What is the vertical distance between $(x,y_1)$ and $(x,y_2)$?

    $|y_2-y_1|$. Vertical distance when $x$-coordinates are equal.

12. Find the perimeter of the triangle with vertices $(0,0)$, $(6,0)$, $(0,8)$.

    $24$. Sides: $6+8+10=24$ using distance formula.

13. Find the distance between $(3,4)$ and $(7,1)$.

    $5$. $\sqrt{(7-3)^2+(1-4)^2}=\sqrt{16+9}=5$

14. Find the perimeter of the triangle with vertices $(0,0)$, $(3,0)$, $(0,4)$.

    $12$. Sides: $3+4+5=12$ using distance formula.

15. Find the perimeter of the polygon with vertices $(0,0)$, $(2,0)$, $(2,2)$, $(0,2)$.

    $8$. Square with side length $2$: perimeter is $4 \times 2=8$.

16. Find the area of the axis-aligned rectangle with corners $(-3,-1)$ and $(2,4)$.

    $25$. Width $5$, height $5$: area is $25$.

17. Identify the method to find a polygon perimeter from vertices listed in order.

    Add distances of consecutive vertices and the last to the first. Calculate distances between ordered consecutive vertices.

18. Find the area of the axis-aligned rectangle with corners $(1,2)$ and $(6,7)$.

    $25$. Width $5$, height $5$: area is $25$.

19. Find the perimeter of the axis-aligned rectangle with corners $(0,0)$ and $(6,8)$.

    $28$. Width $6$, height $8$: perimeter is $2(6+8)=28$.

20. Identify whether segments with slopes $\frac{2}{3}$ and $-\frac{3}{2}$ are perpendicular.

    Yes, because $\frac{2}{3}\cdot\left(-\frac{3}{2}\right)=-1$. Product of slopes equals $-1$, confirming perpendicularity.

21. Find the perimeter of the triangle with vertices $(0,0)$, $(2,0)$, $(2,2)$.

    $4+2\sqrt{2}$. Sides: $2+2+2\sqrt{2}=4+2\sqrt{2}$

22. Find the distance between $(2,-3)$ and $(2,5)$.

    $8$. Vertical distance: $|5-(-3)|=8$

23. What is the midpoint formula for 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 $x$ and $y$ coordinates separately.

24. Find the distance between $(5,-1)$ and $(5,7)$.

    $8$. Vertical distance: $|7-(-1)|=8$

25. What is the slope formula between $(x_1,y_1)$ and $(x_2,y_2)$ (when $x_1\ne x_2$)?

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

26. Find the perimeter of the triangle with vertices $ (0,0) $, $ (2,0) $, $ (0,2) $.

    $4+2\sqrt{2}$. Sides: $2+2+2\sqrt{2}=4+2\sqrt{2}$

27. Find the distance between $(1,2)$ and $(4,6)$.

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

28. Find the perimeter of the axis-aligned rectangle with corners $(1,2)$ and $(6,7)$.

    $20$. Width $5$, height $5$: perimeter is $2(5+5)=20$.

29. Find the perimeter of the triangle with vertices $(0,0)$, $(4,0)$, $(4,3)$.

    $12$. Sides: $4+3+5=12$ using distance formula.

30. What is the distance between $ (0,0) $ and $ (a,b) $?

    $\sqrt{a^2+b^2}$. Direct application of distance formula from origin.

31. Find the area of the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, $(0,2)$.

    $4$. Square with side length $2$: area is $2^2=4$.

32. What is the perimeter formula for a rectangle with side lengths $\ell$ and $w$?

    $P=2\ell+2w$. Sum of all four sides: two lengths plus two widths.

33. Find the distance between $(1,1)$ and $(4,5)$.

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

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

    $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem for coordinate plane distances.

35. What is the area formula for a rectangle with side lengths $\ell$ and $w$?

    $A=\ell w$. Standard rectangle area formula: length times width.

36. Find the distance between $(-2,3)$ and $(4,3)$.

    $6$. Horizontal distance: $|4-(-2)|=6$

37. Find the area of the triangle with vertices $(0,0)$, $(2,0)$, $(2,2)$.

    $2$. Right triangle: $\frac{1}{2} \times 2 \times 2 = 2$

38. Find the perimeter of the square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$.

    $4$. Unit square has perimeter $4 \times 1=4$.

39. Find the midpoint of the segment from $(2,8)$ to $(6,0)$.

    $(4,4)$. Average coordinates: $\left(\frac{2+6}{2},\frac{8+0}{2}\right)$

40. Find the perimeter of the triangle with vertices $ (0,0) $, $ (2,0) $, $ (2,2) $.

    $4+2\sqrt{2}$. Sides: $2+2+2\sqrt{2}=4+2\sqrt{2}$

41. What condition shows two lines are perpendicular using slopes $m_1$ and $m_2$?

    $m_1m_2=-1$. Product of perpendicular slopes equals negative one.

42. Find the slope of the line through $(-2,3)$ and $(4,3)$.

    $0$. Horizontal line has zero slope.

43. Find the area of the triangle with vertices $(0,0)$, $(3,0)$, $(0,4)$.

    $6$. Right triangle: $\frac{1}{2} \times 3 \times 4 = 6$

44. Find the area of the triangle with vertices $(0,0)$, $(2,0)$, $(0,2)$.

    $2$. Right triangle: $\frac{1}{2} \times 2 \times 2 = 2$

45. What condition shows two segments are parallel using slopes $m_1$ and $m_2$?

    $m_1=m_2$. Parallel segments have identical slopes.

46. Find the length of a rectangle diagonal with corners $(0,0)$ and $(6,8)$.

    $10$. Diagonal distance: $\sqrt{6^2+8^2}=\sqrt{100}=10$

47. Find the perimeter of the triangle with vertices $(0,0)$, $(2,0)$, $(0,2)$.

    $4+2\sqrt{2}$. Sides: $2+2+2\sqrt{2}=4+2\sqrt{2}$

48. What is the length of the hypotenuse of a right triangle with legs $a$ and $b$?

    $c=\sqrt{a^2+b^2}$. Pythagorean theorem for right triangle hypotenuse.

49. Find and correct the error: $d=\sqrt{(x_2-x_1)+(y_2-y_1)}$ for distance.

    Correct: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Missing squares in the Pythagorean theorem calculation.

50. Find the area of the triangle with vertices $(0,0)$, $(6,0)$, $(0,8)$.

    $24$. Right triangle: $\frac{1}{2} \times 6 \times 8 = 24$

51. Find the perimeter of the axis-aligned rectangle with corners $(-3,-1)$ and $(2,4)$.

    $20$. Width $5$, height $5$: perimeter is $2(5+5)=20$.

52. What is the midpoint formula for 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 $x$ and $y$ coordinates separately.

53. Find the slope of the line through $(5,-1)$ and $(5,7)$.

    Undefined. Vertical line has undefined slope.

54. What is the horizontal distance between $(x_1,y)$ and $(x_2,y)$?

    $|x_2-x_1|$. Horizontal distance when $y$-coordinates are equal.

55. What is the area of a right triangle with legs of lengths $a$ and $b$?

    $A=\frac{1}{2}ab$. Right triangle area uses legs as base and height.

56. Find the area of the square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$.

    $1$. Unit square has area $1 \times 1=1$.

57. Find the distance between $(0,0)$ and $(3,4)$.

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

58. What is the horizontal distance between $ ($x_1$, y) $ and $ ($x_2$, y) $?

    $|x_2-x_1|$. Horizontal distance when $y$-coordinates are equal.

59. Find the midpoint of the segment from $(-4,1)$ to $(2,7)$.

    $(-1,4)$. Average coordinates: $\left(\frac{-4+2}{2},\frac{1+7}{2}\right)$

60. Identify whether lines through $(0,0)$ to $(2,2)$ and $(1,0)$ to $(3,2)$ are parallel.

    Yes, both slopes are $1$. Both segments have slope $1$, so they're parallel.

61. What is the perimeter of an axis-aligned rectangle with corners $(x_1,y_1)$ and $(x_2,y_2)$?

    $P=2|x_2-x_1|+2|y_2-y_1|$. Sum of all four sides for axis-aligned rectangles.

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