Suppose you want to build a rectangular picture frame where the width is 4 inches less than the length and the diagonal is 4
inches longer than the length. What are the dimensions of the picture frame?
L = length
W = width
d =diagonal
d = √ ( W² + L²)
W = L - 4
d = √ [ ( L - 4 )² + L²]
d = L + 4
L + 4 = √ [ ( L - 4 )² + L²]
Raise both sides to the power of two.
( L + 4 )² = ( L - 4 )² + L²
L²+ 2 L ∙ 4 + 4² = L² - 2 L ∙ 4 + 4² + L²
L²+ 8 L + 16 = L² - 8 L + 16 + L²
Subtract L² + 16 to both sides
8 L = - 8 L + L²
Add 8 L to both sides
8 L + 8 L = - 8 L + L² + 8 L
16 L = L²
Divide both sides by L
16 = L
L = 16 in
W = L - 4 = 16 - 4 = 12
Proof:
d = √ ( W² + L²) = √ ( 16² + 12²) =
√ ( 256 + 144 ) = √ 400 = 20 in
d = L + 4 = 16 + 4
Oh, a picture frame, fancy! Let's solve this picture-perfect puzzle, shall we?
Let's call the length of the frame "L" (because "L" stands for Length, right?) and the width "W" (for Width, of course).
According to our puzzling problem, the width is 4 inches less than the length. So, we can write the equation: W = L - 4.
Now, here comes the interesting part! The diagonal of the frame is a real attention seeker, and it's 4 inches longer than the length. We know that the length, width, and diagonal form a right triangle, so we can use the Pythagorean theorem a^2 + b^2 = c^2, where "c" is the hypotenuse (the diagonal) and "a" and "b" are the other two sides (the length and width, in any order).
Using our equation for the width (W = L - 4), we can rewrite the Pythagorean theorem as: L^2 + (L - 4)^2 = (L + 4)^2.
Now, let's solve this puzzle and find the length (L)! After a little algebraic magic (or should I say, "math-ic"), we find L = 9 inches.
So, the length of the picture frame is 9 inches. Now, substituting this value into our equation for the width, W = L - 4, we find W = 9 - 4 = 5 inches.
Therefore, the dimensions of the picture frame are 9 inches by 5 inches. Ta-da!
Let's solve this step-by-step.
Step 1: Let's assume the length of the picture frame is "x" inches.
Step 2: The width of the picture frame will be 4 inches less than the length, so it will be x - 4 inches.
Step 3: The diagonal of the picture frame is given to be 4 inches longer than the length, so it will be x + 4 inches.
Step 4: Using the Pythagorean theorem, we can find the relationship between the length, width, and diagonal of the rectangle:
length^2 + width^2 = diagonal^2
Step 5: Substitute the values we found into the equation from step 4:
x^2 + (x - 4)^2 = (x + 4)^2
Step 6: Simplify the equation:
x^2 + (x - 4)(x - 4) = (x + 4)(x + 4)
Step 7: Expand both sides of the equation:
x^2 + (x^2 - 8x + 16) = x^2 + 8x + 16
Step 8: Combine like terms:
x^2 + x^2 - 8x + 16 = x^2 + 8x + 16
Step 9: Simplify the equation further:
2x^2 - 8x + 16 = x^2 + 8x + 16
Step 10: Rearrange the equation:
x^2 - 16x = 0
Step 11: Factor out the common term, x:
x(x - 16) = 0
Step 12: Solve for x. Since we can't have a negative length, the only valid solution is x = 0.
Step 13: Since x = 0 is not a valid length, there is no solution for this problem.
Therefore, there are no dimensions for the picture frame that satisfy the given conditions.
To solve this problem, we need to set up equations based on the given information and solve them algebraically.
Let's assume the length of the picture frame be L inches.
1. The width is 4 inches less than the length:
Width = L - 4
2. The diagonal is 4 inches longer than the length:
Diagonal = L + 4
Now, we can use the Pythagorean theorem to find the relationship between the length, width, and diagonal of the rectangle. The Pythagorean theorem states that in a right-angle triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides (length and width).
Applying the Pythagorean theorem, we have:
Length² + Width² = Diagonal²
Substituting the values we found earlier:
L² + (L - 4)² = (L + 4)²
Expanding and simplifying the equation:
L² + L² - 8L + 16 = L² + 8L + 16
Combine like terms:
2L² - 16L + 16 = L² + 8L + 16
Simplifying further:
2L² - 24L = 0
Dividing both sides by 2L to solve for L:
L - 12 = 0
Adding 12 to both sides:
L = 12
Therefore, the length of the picture frame is 12 inches.
Now, we can find the width and diagonal using the equations we derived earlier:
Width = L - 4 = 12 - 4 = 8 inches
Diagonal = L + 4 = 12 + 4 = 16 inches
Thus, the dimensions of the picture frame are 12 inches (length), 8 inches (width), and 16 inches (diagonal).