A walking path is shaped like a rectangle with a width 7 times its length "l". What is a simplified expression for the distance between opposite corners of the walking path?
Did you draw a picture? The distance between opposite corners of the walking path (rectangle, in this case) would be the hypotenuse of a right triangle. Thus, we can use the Pythagorean Theorem: HYP^2 = leg^2 + leg^2
Let x = length; 7x = width, so
HYP^2 = x^2 + (7x)^2
HYP^2 = x^2 + 49x^2
faactor out x^2:
HYP^2 = x^2 * (1+49)
HYP^2 = X^2 * 25 * 2
square root both sides
HYP = 5x * sqrt(2)
To find the distance between opposite corners of the walking path, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's call the length of the rectangle "l". The width of the rectangle is given as 7 times its length, which can be written as 7l.
Using the Pythagorean Theorem, we can set up the following equation:
hypotenuse^2 = length^2 + width^2
In this case, the hypotenuse represents the distance between opposite corners of the walking path.
So, we have:
hypotenuse^2 = l^2 + (7l)^2
Simplifying the equation:
hypotenuse^2 = l^2 + 49l^2
hypotenuse^2 = 50l^2
To find a simplified expression for the distance between opposite corners, we can take the square root of both sides:
hypotenuse = sqrt(50l^2)
Since we want a simplified expression, we can also simplify the square root:
hypotenuse = sqrt(2*5*5l^2)
hypotenuse = 5l*sqrt(2)
Therefore, a simplified expression for the distance between opposite corners of the walking path is 5l times the square root of 2.
To find the distance between opposite corners of a rectangle, we can use the Pythagorean theorem.
Let's consider the length of the rectangle as "l".
According to the given information, the width of the rectangle is 7 times its length, which means the width is 7l.
Now, we can draw a right-angled triangle using the length and width of the rectangle as its two sides.
The diagonal or the distance between opposite corners of the rectangle is the hypotenuse of this right-angled triangle.
According to the Pythagorean theorem, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
In this case, the length (l) is one side, and the width (7l) is the other side.
So, the simplified expression for the distance between opposite corners of the walking path is:
√(l^2 + (7l)^2)
Simplifying this expression further, we get:
√(l^2 + 49l^2)
√(50l^2)
√(50) * √(l^2)
√(2 * 5^2) * l
5√2 * l
Thus, the simplified expression for the distance between opposite corners of the walking path is 5√2 times the length (l).