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: Calculate distance from origin

Use distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
For point $(-4,5)$ and origin $(0,0)$:
$d=\sqrt{(-4-0)^2+(5-0)^2}=\sqrt{(-4)^2+5^2}$

Step2: Simplify the expression

$\sqrt{16+25}=\sqrt{41}$

Step3: Compare to radius 6

$\sqrt{41}\approx6.403$, which is $>6$

Answer:

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