In the xy-plane, c and d are circles centered at the origin with radii p 17 and p 5, respectively. quantity a: the number of points (a; b) on circle c where both a and b are integers quantity b: the number of points (a; b) on circle d where both a and b are integers a. quantity a is greater. b. quantity b is greater. c. the two quantities are equal. d. the relationship cannot be determined from the information given.

Quantity A is greater

Step-by-step explanation:

Well, to solve this we'll do it by parts.

Quantity A) 809 coordinate points

Since it's centered at the origin we can write the circle equation as

For now, we set apart the following points (0,17) (0,-17), (17,0) and (-17,0) because those points already belong to the circumference and circle.

Now we're going to enlist the cartesian points. The parameter is the Radius. No point can be outside the circle, i.e. with a distance longer than 289 units.

Doing it with the Quadrant I. We'll enlist them. And test by squaring

(x,y) x²+y²≤289

(0,0) Origin

(1,1); ...(1,16) (1,16)∈ for 1²+16² ≤289

(1,1); ...(1,16) = 16 points

(2,1)....(2,16) =16 points (7,1)...(7,15) =15 points (12,1)...(12,12)=12 points

(3,1)....(3,16) =16 points (8,1)....(8,15) =15 points (13,1)...(13,10)= 10 points

(4,1)....(4,16) =16 points (9,1)...(9,14) =14 points (14,1)...(14,9)= 9 points

(5,1)....(5,15) =15 points (10,1)...(10,13) =13 points (15,1)...(15,8) =8 points

(6,1)...(6,15) =15 points (11,1)...(11,12)= 12 points (16,1)...(16,5) =5 points

So

16+16+16+15+15+15+15+14+13+12+12+10+9+8+5=

48+60+27+24+27= 186 coordinate points for the Quadrant I

186*4=744 coordinate points for all four Quadrants

y axis points: (0,1)..(0,16) =16 points (0,-1)...(0,-16)=16 points

x-axis points (1,0)...(16,0)=16 points (-1,0)...(-16,0)=16 points

Origin (0,0)= 1 coordinate point

Notice that I'm only counting the points on the circle not on the circumference, as the question asks.

Adding it all up:

744 +16+16+16+16+1=809 coordinate points

2) Quantity B: 77 coordinate points

Similarly:

x²+y²=5²

x²+y²=25

Points on the y-axis: 8 coordinate points. Points on x-axis: 8 coordinate points

(1,1)...(1,4) =4 points (2,1)...(2,4) =4 points (3,1)..(3,4)=4 points (4,1)...(4,3) =3 points

In Quadrant I = 15

In 4 quadrants = 60 coordinates points

60 coordinates points +4+4+4+4+1=77 coordinate points

Final comments:

We could solve it this only estimating. Just based on the radii of those circles. Since the question does not ask any quantity, but just the relationship. This could be a time saver way.

parallel and congruent

step-by-step explanation:

the movement of translation is linear.   these have the same translation because you are using the same rule for each point of the triangle so you will see the same slope being used to get from a point on the triangle to its image.

same slope!   you should think parallel when you see those 2 word together (well unless it's the same line but it can't be because we are coming from different points here)

congruent! because each point is being moved the same distance.