can someone correct this for me thanks.
The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 4 cm, what are the dimensions (the length and the width) of the rectangle?
My answer: x = 2.284cm wide and 3.284cm long.
correct.
To solve this problem, we can use the Pythagorean theorem. According to the problem statement, the length of the rectangle is 1 cm longer than its width.
Let's assume the width of the rectangle is 'x' cm. Then, the length of the rectangle would be 'x + 1' cm.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (in this case, the diagonal) is equal to the sum of the squares of the other two sides (in this case, the length and width of the rectangle).
So, if we consider the width as the base and the length as the height, the diagonal can be considered as the hypotenuse in a right-angled triangle.
Applying the Pythagorean theorem, we have:
(diagonal)^2 = (width)^2 + (length)^2
4^2 = x^2 + (x + 1)^2
16 = x^2 + (x^2 + 2x + 1)
16 = 2x^2 + 2x + 1
Rearranging the equation, we have:
2x^2 + 2x - 15 = 0
Now, we can solve this quadratic equation to find the value of 'x':
Using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Here, a = 2, b = 2, and c = -15.
Substituting the values, we get:
x = (-2 ± sqrt(2^2 - 4*2*(-15))) / (2*2)
x = (-2 ± sqrt(4 + 120)) / 4
x = (-2 ± sqrt(124)) / 4
x = (-2 ± 2sqrt(31)) / 4
x = (-1 ± sqrt(31)) / 2
So, the possible values of 'x' are (-1 + sqrt(31)) / 2 and (-1 - sqrt(31)) / 2.
However, since the width of the rectangle cannot be negative, we discard the negative value.
Therefore, the width of the rectangle is (sqrt(31) - 1) / 2 cm.
Substituting this into the expression for the length, we can find the length of the rectangle:
Length = Width + 1
Length = (sqrt(31) - 1) / 2 + 1
Length = (sqrt(31) - 1 + 2) / 2
Length = (sqrt(31) + 1) / 2
So, the dimensions of the rectangle are approximately:
Width ≈ (sqrt(31) - 1) / 2 cm
Length ≈ (sqrt(31) + 1) / 2 cm
Rounding to three decimal places, we have:
Width ≈ 2.284 cm
Length ≈ 3.284 cm
Therefore, your answer is correct: The width of the rectangle is approximately 2.284 cm, and the length is approximately 3.284 cm.