1.find the distance between the points (-1,2) and (3,4).express your answer in simplest radical form.
3.a circle with center C(4,-2) passes through the point A(1,3).does the point B(8,-2) lie inside the circle?prove your answer.
1.
change in x = 4
change in y = 2
so
hypotenuse = sqrt (16 + 4) = sqrt (20) = sqrt(2*2*5) = 2 sqrt 5
How far is A from the center? (see last problem for how to calculate distance from C to A)
then
How far is B from the center ? ( same deal)
if B is further from the center than A, then B is outside the circle.
the radius is ...√[(3 - -2)^2 + (1 - 4)^2]
the distance to B is ... 8 - 4
1. Well, I don't want to go all squiggly on you, but I'd say let's get some mathematical clowning done here. Let me grab my rubber nose and oversized shoes. Now, the distance between two points can be found using the good ol' distance formula. So, clown math time! The distance between (-1,2) and (3,4) is equal to the square root of [(3 - (-1))^2 + (4 - 2)^2]. Simplify that funky expression and you'll get your answer.
3. Alright, let's get serious for a moment. We need to determine whether the point B(8,-2) lies inside the circle with center C(4,-2) and passing through point A(1,3). The distance between the centers of the circle and point B is equal to the square root of [(8 - 4)^2 + (-2 - (-2))^2]. Now, if this distance is greater than the radius of the circle (which is the distance between the center and point A), then B is outside the circle. But if it's less than or equal to the radius, then B lies inside the circle. Do the math and you'll get your answer, my friend. Enjoy the circus of geometry!
To find the distance between two points, we can use the distance formula, which is derived from the Pythagorean theorem. The formula is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For the first question, we need to find the distance between the points (-1, 2) and (3, 4).
Using the formula, we have:
Distance = sqrt((3 - (-1))^2 + (4 - 2)^2)
= sqrt((3 + 1)^2 + (4 - 2)^2)
= sqrt(4^2 + 2^2)
= sqrt(16 + 4)
= sqrt(20)
= 2sqrt(5)
So, the distance between the points (-1, 2) and (3, 4) is 2sqrt(5) in simplest radical form.
For the second question, we need to determine whether the point B(8, -2) lies inside the circle with center C(4, -2) and passing through the point A(1, 3).
To do this, we can calculate the distance between the center of the circle and point B. If the distance is less than the radius of the circle, then the point lies inside the circle.
The distance formula is:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the formula, we have:
Distance = sqrt((8 - 4)^2 + (-2 - (-2))^2)
= sqrt((4)^2 + (0)^2)
= sqrt(16 + 0)
= sqrt(16)
= 4
The distance between the center C(4, -2) and point B(8, -2) is 4.
Since the radius is not provided, if we assume the radius of the circle is 4 (which is the distance between C and B), then B lies on the circle.
If the radius is greater than 4, point B would lie outside the circle. Otherwise, if the radius is less than 4, point B would be inside the circle.
To prove this, you can calculate the distance between point A and the center C, and if it is equal to the radius, it confirms that point A lies on the circle.