Video: Determining Whether a Point Is within a Circular Region given Its Centre and Radius

A radar is located at the point 𝐴 (−2, −4) covering a circular region with a radius of 27 length units. Can the radar observe a ship at the point 𝐵 (10, 0)?

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Video Transcript

A radar is located at the point 𝐴: negative two, negative four covering a circular region with a radius of 27 length units. Can the radar observe a ship at the point 𝐵: 10, zero?

The distance between any two points can be calculated using the formula: the square root of 𝑥 two minus 𝑥 one all squared plus 𝑦 two minus 𝑦 one all squared. In this case, we need to calculate the distance 𝐴𝐵 from the point 𝐴, where the radar is located, to the ship at point 𝐵 and determine whether it is within the circular region that the radar covers.

Substituting in the coordinates gives us an equation: the square root of 10 minus negative two all squared plus zero minus negative four all squared. As 10 minus negative two is 12 and zero minus negative four is four, we are left with 12 squared plus four squared. 12 squared is equal to 144. Four squared is equal to 16. And the square root of 160 is equal to 12.65.

As 12.65 is less than 27, we can say yes, the radar can observe the ship at point 𝐵: 10, zero. Had the distance 𝐴𝐵 been greater than 27, the radar wouldn’t have been able to observe the ship.

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