Question

determining whether a point lies on a circle
the local pizza restaurant offers free delivery if you live within a 6 - mile radius of the restaurant. the restaurant is located at the origin. adas house is 4 miles west and 5 miles north of the restaurant at point (-4, 5). does adas house lie on or within the circle representing the area that gets free delivery?
no, the distance from the restaurant to adas house is $sqrt{41}$ miles, which is greater than the 6 - mile maximum radius.
yes, the distance from the restaurant to adas house is exactly 6 miles.
no, the distance from the restaurant to adas house is 7 miles, which is greater than the 6 - mile maximum radius.
yes, the distance from the restaurant to adas house is $sqrt{20}$ miles, which is less than the needed 6 - mile radius.

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here the restaurant is at $(0,0)$ and Ada's house is at $(-4,5)$, so $x_1 = 0,y_1=0,x_2=-4,y_2 = 5$.

Step2: Calculate the distance

$d=\sqrt{(-4 - 0)^2+(5 - 0)^2}=\sqrt{(-4)^2+5^2}=\sqrt{16 + 25}=\sqrt{41}\approx6.4$.

Step3: Compare with radius

Since $\sqrt{41}\approx6.4>6$, Ada's house is outside the 6 - mile radius.

Answer:

No, the distance from the restaurant to Ada's house is $\sqrt{41}$ miles, which is greater than the 6 - mile maximum radius.