Distance and the Pythagorean Theorem Quick Check
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Question
Use the image to answer the question.
A graph shows quadrant 1 with the x-axis and y-axis ranging from 0 to positive 5. Data points marked are point A at left parenthesis 1 comma 2 right parenthesis, point B at left parenthesis 3 comma 2 right parenthesis, point C at left parenthesis 3 comma 4 right parenthesis, and point D at left parenthesis 1 comma 4 right parenthesis. A line joins points A to B, B to C, C to D, and D to A, forming a rectangle.
Apply the Pythagorean Theorem to find the length of the diagonal of the square. Round your answer to the nearest hundredth, if necessary.
(1 point)
Responses
8
8
4.24
4.24
2.83
2.83
4
The length of the diagonal of the square can be found using the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, the diagonal of the square is the hypotenuse of the right triangle formed by points A, B, and C. The lengths of the other two sides are the horizontal distance between points A and C, and the vertical distance between points A and C.
The horizontal distance between points A and C is 3 - 1 = 2 units.
The vertical distance between points A and C is 4 - 2 = 2 units.
Using the Pythagorean Theorem, the length of the diagonal can be found as follows:
Diagonal^2 = (horizontal distance)^2 + (vertical distance)^2
Diagonal^2 = 2^2 + 2^2
Diagonal^2 = 4 + 4
Diagonal^2 = 8
Taking the square root of both sides to solve for the diagonal:
Diagonal = √8
Rounding √8 to the nearest hundredth, we get approximately 2.83.
Therefore, the correct answer is 2.83.