The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 5 cm, what are the dimensions of the rectangle in centimeters? Enter your answers in the blanks. Enter only the numeric values rounded to the nearest tenth.
Let x be the short side, then the long side is (x+1).
By Pythagoras theorem, the diagonal is
√(x²+(x+1)²)
=&radic(2x²+2x+1)=5
Square both sides to get
2x²+2x+1 = 5
2(x²+x-2)=0
2(x+2)(x-1)=0
Solve for x and reject the root that is negative.
Sorry, there was a mistake in the above solution. Here's the corrected version.
√(x²+(x+1)²)
=√(2x²+2x+1)=5
Square both sides to get
2x²+2x+1 = 25
2(x²+x-12)=0
2(x-3)(x+4)=0
Solve for x and reject the root that is negative.
To find the dimensions of the rectangle, we can use the Pythagorean theorem. Let's denote the width as "w" and the length as "l".
According to the given information, the length is 1 cm longer than the width. Therefore, we can express the length as:
l = w + 1
Now, let's apply the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides (width and length):
w^2 + (w + 1)^2 = 5^2
Simplifying the equation:
w^2 + (w^2 + 2w + 1) = 25
2w^2 + 2w - 24 = 0
Now, we can solve this quadratic equation to find the value of "w". We can use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 2, and c = -24.
w = (-2 ± √(2^2 - 4*2*(-24))) / (2*2)
w = (-2 ± √(4 + 192)) / 4
w = (-2 ± √196) / 4
w = (-2 ± 14) / 4
Thus, we have two possible values for "w":
w₁ = (-2 + 14) / 4 = 12 / 4 = 3
w₂ = (-2 - 14) / 4 = -16 / 4 = -4
Since the width cannot be negative, we discard the negative solution.
Therefore, the width of the rectangle is 3 cm.
Now, we can substitute this value back into the expression for the length:
l = w + 1
l = 3 + 1
l = 4
So, the dimensions of the rectangle are width = 3 cm and length = 4 cm.