Use the image to answer the question.
A graph shows four quadrants with the x-axis and y-axis both ranging from negative 5 to positive 5 in increments of 1. Unlabeled data points marked are at left parenthesis negative 2 comma negative 1 right parenthesis, and at left parenthesis 2 comma 4 right parenthesis. A straight line joins the two lines with a positive slope.
Apply the Pythagorean Theorem to find the length between the two points. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
6:40
6:40
9
9
41
41
5.57
To find the distance between the two points, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the two points form the two sides of a right triangle, with the distance between them being the hypotenuse. The horizontal distance between the points is 2 - (-2) = 4. The vertical distance between the points is 4 - (-1) = 5.
Using the Pythagorean theorem, we can find the length of the hypotenuse:
Length^2 = (horizontal distance)^2 + (vertical distance)^2
Length^2 = 4^2 + 5^2
Length^2 = 16 + 25
Length^2 = 41
Taking the square root of both sides, we find that the length of the hypotenuse (or the distance between the points) is the square root of 41, which is approximately 6.40. Rounding to the nearest hundredth, the length between the two points is 6.40.
To apply the Pythagorean Theorem, we need to calculate the length of the line joining the two points.
Let's label the first point as A (-2, -1) and the second point as B (2, 4).
The length of the line AB can be found using the formula:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of point A (-2, -1) and point B (2, 4) into the formula:
AB = sqrt((2 - (-2))^2 + (4 - (-1))^2)
= sqrt((4 + 2)^2 + (4 + 1)^2)
= sqrt(6^2 + 5^2)
= sqrt(36 + 25)
= sqrt(61)
Rounded to the nearest hundredth, the length of the line AB is approximately 7.81.
To find the length between the two points on the graph, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, we have two points: (-2, -1) and (2, 4). The x-coordinates of these points are -2 and 2, and the y-coordinates are -1 and 4, respectively. We can imagine a right triangle formed by connecting these two points with a straight line.
Using the Pythagorean theorem, we can find the length of the hypotenuse (the straight line joining the two points):
Length = √((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we get:
Length = √((2 - (-2))^2 + (4 - (-1))^2)
= √((4)^2 + (5)^2)
= √(16 + 25)
= √41
Therefore, the length between the two points is √41. If we round it to the nearest hundredth, it becomes 5.57. So the answer is 5.57.