the diagonal of a rectangular room is 13ft long. one wall measures 7ft longer than the adjacent wall. what are the dimensions of the room
13=(x+7)+x
x=3
So i think it would be 10X 3?
The diagonal forms 2 rt triangles: it is the hyp of the triangles.
Width = X ft.
Length = (x+7) ft.
x^2 + (x+7)^2 = (13)^2,
x^2 + x^2 + 14x + 49 = 169,
2x^2 + 14x - 120 = 0,
Divide both sides by 2:
x^2 + 7x - 60 = 0,
(x-5)(x+12) = 0,
x-5 = 0,
x = 5.
x+12 = 0,
x = -12.
Select + value of X:
X = 5 ft = Width.
X + 7 = 12 ft = Length.
Let's assume the length of the shorter adjacent wall is x feet.
Since the longer wall is 7 feet longer than the adjacent wall, the length of the longer wall would be (x + 7) feet.
According to the given information, the diagonal of the rectangular room is 13 feet long.
We can use the Pythagorean theorem to find the third side, which is the height (h), of the room.
The Pythagorean theorem states that the square of the hypotenuse (the diagonal) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is 13 feet, and the other two sides are the lengths of the adjacent sides.
So, we can write the equation as:
x^2 + (x + 7)^2 = 13^2
Simplifying the equation:
x^2 + x^2 + 14x + 49 = 169
Combining like terms:
2x^2 + 14x + 49 = 169
Rearranging the equation:
2x^2 + 14x - 120 = 0
Simplifying the equation further by dividing by 2:
x^2 + 7x - 60 = 0
We can now factor the equation:
(x - 5)(x + 12) = 0
Setting each factor equal to zero:
x - 5 = 0 or x + 12 = 0
x = 5 or x = -12
Since the length of the wall cannot be negative, we can discard x = -12.
Therefore, the length of the shorter adjacent wall is 5 feet.
The length of the longer wall would be (x + 7) = (5 + 7) = 12 feet.
Hence, the dimensions of the room are 5 feet by 12 feet.
To find the dimensions of the room, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
Let's assume that the length of the shorter wall is "x" feet. Since the longer wall is 7 feet longer than the adjacent wall, its length will be "x + 7" feet.
According to the Pythagorean theorem, we have:
(x^2) + ((x + 7)^2) = (13^2)
Simplifying this equation, we get:
x^2 + (x^2 + 14x + 49) = 169
Combining like terms:
2x^2 + 14x + 49 = 169
Rearranging the equation:
2x^2 + 14x - 120 = 0
Now, we can solve this quadratic equation for "x" using factoring, completing the square, or the quadratic formula. Factoring is the simplest method in this case.
Factoring the equation, we get:
2(x - 4)(x + 15) = 0
Setting each factor equal to zero, we have two possible solutions:
x - 4 = 0 or x + 15 = 0
Solving for "x" in each case, we find:
x = 4 or x = -15
Since the dimensions of a room cannot be negative, we discard the second solution. Therefore, the length of the shorter wall (x) is 4 feet.
The longer wall is the adjacent wall plus 7 feet, so its length is:
x + 7 = 4 + 7 = 11 feet
Hence, the dimensions of the room are 4 feet by 11 feet.