The width of a rectangle is 8 less than twice its length. If the area of the rectangle is 172 cm^2, what is the length of the diagonal?
W = 2L - 8
W * L = 172
Substitute 2L-8 for W in the second equation and solve for L. Insert that value into the first equation to solve for W. Check by putting both values into the second equation.
Then use Pythagorean theorem.
W^2 + L^2 = diagonal^2
Step 1: Set up the equation for the width and length of the rectangle.
Let's assume the length of the rectangle is x cm.
According to the given information, the width is 8 less than twice the length.
So, the width can be represented as 2x - 8 cm.
Step 2: Find the area of the rectangle using the formula: Area = Length x Width.
As the area of the rectangle is given as 172 cm^2, we can set up the equation:
x * (2x - 8) = 172.
Step 3: Solve the quadratic equation to find the value of x.
Expanding the equation:
2x^2 - 8x = 172.
Rearranging the equation:
2x^2 - 8x - 172 = 0.
Simplifying the equation:
x^2 - 4x - 86 = 0.
No factorization is possible in this case, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values:
x = (-(-4) ± √((-4)^2 - 4 * 1 * -86)) / (2 * 1).
Simplifying:
x = (4 ± √(16 + 344)) / 2.
x = (4 ± √360) / 2.
Step 4: Calculate the possible values of x.
Using the positive square root:
x = (4 + √360) / 2.
Simplifying:
x = (4 + 6√10) / 2.
x = 2 + 3√10.
Therefore, the length of the rectangle is 2 + 3√10 cm.
Step 5: Find the diagonal of the rectangle using the Pythagorean theorem.
The diagonal of a rectangle can be calculated using the formula:
Diagonal^2 = Length^2 + Width^2.
Substituting the values:
Diagonal^2 = (2 + 3√10)^2 + (2(2 + 3√10) - 8)^2.
Simplifying:
Diagonal^2 = (2 + 3√10)^2 + (4 + 6√10 - 8)^2.
Diagonal^2 = (2 + 3√10)^2 + (6√10 - 4)^2.
Diagonal^2 = (2 + 3√10)^2 + (6√10 - 4)^2.
Diagonal^2 = 4 + 12√10 + 9(10) + 36(10) - (48)(√10) + 16 + 36 - (48)(√10).
Diagonal^2 = 4 + 36 + 90 + 360 - 96√10 + 16 + 36 - 96√10.
Diagonal^2 = 186 + 426 - 192√10.
Diagonal^2 = 612 - 192√10.
Therefore, the square of the diagonal is 612 - 192√10 cm^2.
To find the length of the diagonal of a rectangle, we need to know the length and width of the rectangle. From the given information, we can set up two equations to find the length and width.
Let's assume that the length of the rectangle is represented by "L" and the width is represented by "W".
Based on the problem statement, we can write the first equation:
W = 2L - 8
This equation indicates that the width "W" is 8 less than twice the length "L".
The second equation comes from the fact that the area of a rectangle is equal to its length multiplied by its width:
Area = Length × Width
172 cm^2 = L × W
Now, substitute the value of "W" from the first equation into the second equation:
172 cm^2 = L × (2L - 8)
Simplify the equation:
172 cm^2 = 2L^2 - 8L
Rearrange the equation into a quadratic form:
2L^2 - 8L - 172 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring the equation, we get:
2(L - 14)(L + 6) = 0
This gives us two possible solutions for L:
L - 14 = 0 or L + 6 = 0
Solving each equation, we find:
L = 14 or L = -6
Since the length of a rectangle cannot be negative, we discard the solution L = -6.
Therefore, the length of the rectangle is L = 14 cm.
Now, to find the width, substitute the value of L back into the first equation:
W = 2L - 8
W = 2(14) - 8
W = 28 - 8
W = 20 cm
Now that we have the length (L = 14 cm) and width (W = 20 cm) of the rectangle, we can calculate the length of the diagonal using the Pythagorean theorem.
In a rectangle, the diagonal, length, and width form a right triangle. So, using the Pythagorean theorem, we have:
diagonal^2 = length^2 + width^2
Substituting the values, we get:
diagonal^2 = 14^2 + 20^2
diagonal^2 = 196 + 400
diagonal^2 = 596
To find the length of the diagonal, we take the square root of both sides:
diagonal = √596
Using a calculator, we find that the length of the diagonal is approximately 24.41 cm.