Question 756·Medium·Math

0 / 2000

A circular stage is shown on an $xy$ -coordinate grid. The center of the stage is at $C(2,-1)$ , and a point on the edge of the stage is at $P(8,5)$ .

Which choice is the area of the stage, in square units, in terms of $\pi$ ?

A circle’s radius runs from its center to a point on its edge. On a coordinate grid, the horizontal and vertical changes form a right triangle, so the sum of their squares gives the radius squared. Type that sum into Desmos, then use $A=\pi r^2$ for area. Don’t halve a center-to-edge segment; only a full diameter gets halved.

Hints

Hint 1
A radius joins the center to the edge, while a diameter runs from one edge to the other through the center. The points have different jobs here. What kind of segment connects $C$ and $P$ ?
Hint 2
The horizontal and vertical changes form perpendicular sides of a right triangle. Subtract $x$ from $x$ and $y$ from $y$ . What happens when you subtract the center’s negative $y$ -coordinate?
Hint 3
The Pythagorean theorem says the squares of those two changes add to the squared radius, $r^2$ . Circle area uses $r^2$ , so you don’t need to find or round the radius itself.

Step-by-step

Find the squared radius, then the area
Step 1
Identify the radius
The figure draws $\overline{CP}$ from the center $C$ to the edge at $P$ . A radius runs center to edge, so $CP$ is already one radius, not a diameter. Don’t halve it: a diameter runs all the way across the circle.
Step 2
Find the coordinate changes
From $C(2,-1)$ to $P(8,5)$ , subtract matching coordinates. The horizontal change is $8-2=6$ , and the vertical change is $5-(-1)=6$ . Subtracting $-1$ means the segment rises $6$ units, not $4$ .
Step 3
Calculate the squared radius
The horizontal and vertical changes meet at a right angle. By the Pythagorean theorem, their squares add to $r^2$ . Type $(8-2)^2+(5-(-1))^2$ into Desmos. It prints $72$ , so the squared radius is $r^2=72$ .
Step 4
Find the stage’s area
The stage’s area is the space inside the circle, so use $A=\pi r^2$ . Substitute the squared radius:
$A=\pi(72)=72\pi.$
The stage has an area of $72\pi$ square units. Choice C.

Question 756·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this