Question

Which point is closest to the origin?

A) (-5,-7)
B) (8,-4)
C) (6,6)
D) (9,0)

Answer 1

The point closest to the origin is found by calculating the distance of each point to the origin using the distance formula. After comparing distances, point C (6, 6) is the closest with the shortest distance to the origin.

Step-by-step explanation:

To determine which point is closest to the origin, we calculate the distance for each point from the origin using the distance formula √(x² + y²), where (x, y) are the coordinates of the point.

• For point A (-5, -7), the distance is √((-5)² + (-7)²) = √(25 + 49) = √74.
• For point B (8, -4), the distance is √((8)² + (-4)²) = √(64 + 16) = √80.
• For point C (6, 6), the distance is √((6)² + (6)²) = √(36 + 36) = √72.
• For point D (9, 0), the distance is √((9)² + (0)²) = √(81) = 9.

Comparing the distances, point C has the shortest distance to the origin, and therefore, point C (6,6) is closest to the origin.

Answer 2

`(6,\, 6)`.

Step-by-step explanation:

The origin of a cartesian plane is at
`(0,\, 0)`.

The distance between two points
`(x_(0),\, y_(0))` and
`(x_(1),\, y_(1))` on a cartesian plane is:

`\displaystyle \sqrt{(x_(1) - x_(0))^(2) + (y_(1) - y_(0))^(2)}`.

Using this equation, find the distance between each option and the origin.

`\sqrt{((-5) - 0)^(2) + ((-7) - 0)^(2)} = √(74)`.

`\sqrt{((8) - 0)^(2) + ((-4) - 0)^(2)} = √(80)`.

`\sqrt{((6) - 0)^(2) + ((6) - 0)^(2)} = √(72)`.

`\sqrt{((9) - 0)^(2) + ((0) - 0)^(2)} = √(81)`.

Hence, among the options, the distance between the point
`(6,\, 6)` and the origin is the smallest:
`√(72)`.