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
4
4
8
8
2.83
2.83
4.24
To find the length of the diagonal of the square, we need to calculate the distance between opposite corners of the square. Using the Pythagorean Theorem, we can find this distance:
The distance between points A and C is the height of the rectangle, which is 4 - 2 = 2 units.
The distance between points B and D is the width of the rectangle, which is 3 - 1 = 2 units.
Using the Pythagorean Theorem, we have:
d^2 = 2^2 + 2^2
d^2 = 4 + 4
d^2 = 8
Taking the square root of both sides, we find:
d ≈ √8
d ≈ 2.83
Therefore, the length of the diagonal of the square is approximately 2.83 units (rounded to the nearest hundredth).
To find the length of the diagonal of the square, we need to calculate the distance between two opposite corners. In this case, the opposite corners are points A and C.
We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, we can consider the x and y coordinates of points A and C as the sides of a right triangle. Using the formula, we have:
- the length of side A to C (the hypotenuse) = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we have:
- x1 = 1, y1 = 2 (coordinates of point A)
- x2 = 3, y2 = 4 (coordinates of point C)
Applying the formula, we get:
- (length of diagonal)^2 = (3 - 1)^2 + (4 - 2)^2
- (length of diagonal)^2 = 2^2 + 2^2
- (length of diagonal)^2 = 4 + 4
- (length of diagonal)^2 = 8
To find the length of the diagonal, we need to take the square root of both sides:
- length of diagonal = sqrt(8)
Rounding to the nearest hundredth, we have:
- length of diagonal ≈ 2.83
Therefore, the correct answer is 2.83.
To find the length of the diagonal of the square, we can use the Pythagorean Theorem. The Pythagorean Theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the two sides of the square form a right triangle, with one side being the length of the rectangle (which is 2) and the other side being the width of the rectangle (which is 3). Let's calculate it step-by-step:
1. Square the length of the rectangle: 2^2 = 4.
2. Square the width of the rectangle: 3^2 = 9.
3. Add the two squares together: 4 + 9 = 13.
4. Take the square root of the sum to find the length of the diagonal: √13 ≈ 3.61.
Therefore, the length of the diagonal of the square, rounded to the nearest hundredth, is approximately 3.61.