MHB Which point is farther from the origin?

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To determine which point is farther from the origin, the distance formula can be applied to both points (3, -2) and (4, 1/2). The squared distance from the origin for point (3, -2) is calculated as 13, while for point (4, 1/2), it is 65/4. Comparing these values shows that 65/4 is greater than 13, indicating that (4, 1/2) is farther from the origin. The discussion emphasizes the efficiency of comparing squared distances rather than calculating the actual distances. Ultimately, point (4, 1/2) is confirmed to be the farther point from the origin.
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Which point is farther from the origin?

(3, -2) or (4, 1/2)?

I know the origin is the point (0, 0). This is the location on the xy-plane where the x-axis and y-axis meet. Can the distance formula normally used to calculate how far points on the xy-plane are from each other be applied here? If so, how is the distance formula used in this example?
 
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What you want to do is calculate the distance of both points from the origin, and then compare these distances...or in fact, you could compare the square of the distances. So, we could write:

$$d_1^2=(3-0)^2+(-2-0)^2=13$$

$$d_2^2=(4-0)^2+\left(\frac{1}{2}-0\right)^2=\frac{65}{4}$$

Since $$\frac{65}{4}>\frac{52}{4}=13$$, we find the second given point is farther from the origin. :D
 
MarkFL said:
What you want to do is calculate the distance of both points from the origin, and then compare these distances...or in fact, you could compare the square of the distances. So, we could write:

$$d_1^2=(3-0)^2+(-2-0)^2=13$$

$$d_2^2=(4-0)^2+\left(\frac{1}{2}-0\right)^2=\frac{65}{4}$$

Since $$\frac{65}{4}>\frac{52}{4}=13$$, we find the second given point is farther from the origin. :D

Your way is much faster.
 
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