A playground is shaped like a rectangle with a width 5 times its length (l). What is a simplified expression for the distance between opposite corners of the playground?
I can't copy the multi-choice answers but can you help?
Length is l. Width is w. So what is the expression for the relationship between l and w?
Once you get this, can you come up with the expression for the distance between the two corners?
Yes, I can help! To find the distance between opposite corners of the playground, we can use the Pythagorean theorem.
Let's assume the length of the rectangle is "l". Since the width is 5 times the length, the width would be 5l.
The diagonal of the rectangle forms a right triangle with the length (l) as one side and the width (5l) as the other side.
According to the Pythagorean theorem, the square of the length of the hypotenuse (the diagonal) is equal to the sum of the squares of the lengths of the other two sides.
Using the Pythagorean theorem, we have:
(diagonal)^2 = (length)^2 + (width)^2
(diagonal)^2 = (l)^2 + (5l)^2
(diagonal)^2 = l^2 + 25l^2
(diagonal)^2 = 26l^2
To simplify this expression, we can take the square root of both sides:
diagonal = √(26l^2)
So, the simplified expression for the distance between opposite corners of the playground is √(26l^2).
Sure! To find the distance between opposite corners of the playground, we can use the Pythagorean Theorem. The Pythagorean Theorem 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 other two sides.
In this case, we can consider the width of the rectangle as one side of the right triangle, the length as the other side, and the distance between opposite corners as the hypotenuse. Let's say the length of the rectangle is "l", then the width would be 5 times the length, which is 5l.
Now, using the Pythagorean Theorem, we can write the equation:
c² = a² + b²
Where:
c is the distance between opposite corners (our hypotenuse)
a is the width (5l)
b is the length (l)
Substituting the values into the equation, we get:
c² = (5l)² + l²
Expanding and simplifying:
c² = 25l² + l²
c² = 26l²
To get the simplified expression for the distance between opposite corners, we can take the square root of both sides of the equation:
c = √(26l²)
Therefore, the simplified expression for the distance between opposite corners of the playground is √(26l²).