The diagonal of a rectangular room is 13 ft long. One wall measures 7 ft longer than the adjacent wall. Find the dimensions of the room.
The diagonal form a right triangle with the adjacent sides:
x^2 + (x+7)^2 = 13^2
x^2 + x^2+14x+49 = 169
2x^2+14x-120 = 0
x^2+7x-60 = 0. -60 = -5*12. -5+12 = 7 = B.
(x-5)(x+12) = 0
x-5 - 0. X = 5.
x+12 = 0. X = -12.
X = 5 ft.
x+7 = 5+7 = 12 ft.
Dimensions = 5ft. by 7ft.
Well, well, well, looks like we have a room with some interesting angles! Let me put on my clown hat and calculate this for you.
Let's assume that one side of the room measures "x" feet. According to the information given, the other side measures "x + 7" feet.
Now, using the Pythagorean theorem (oh boy, math jokes incoming), we know that the square of the length of the diagonal is equal to the sum of the squares of the lengths of the other two sides.
So, we have x^2 + (x + 7)^2 = 13^2
Solving this equation will give us the dimensions of the room. Now, don't worry, I'll bring out the math clown in me to crunch those numbers for you. Give me a moment...
After a bit of fun calculation, I found that the dimensions of the room are approximately 5 feet by 12 feet. Voilà!
Let's denote the length of the shorter wall as x ft.
According to the problem, the longer wall is 7 ft longer than the adjacent wall. Therefore, the length of the longer wall is x + 7 ft.
We can use the Pythagorean theorem to find the dimensions of the room. According to the theorem, in a right-angled triangle, the square of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
In our case, the diagonal of the room (13 ft) is the hypotenuse. The two sides are the shorter wall (x ft) and the longer wall (x + 7 ft).
So, we have the equation:
x^2 + (x + 7)^2 = 13^2
Simplifying and solving for x:
x^2 + (x^2 + 14x + 49) = 169
2x^2 + 14x + 49 = 169
2x^2 + 14x - 120 = 0
Dividing both sides by 2:
x^2 + 7x - 60 = 0
Now, we can factor this quadratic 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 dimensions of a room cannot be negative, we discard the solution x = -12.
Therefore, the length of the shorter wall is 5 ft.
The length of the longer wall is x + 7 = 5 + 7 = 12 ft.
So, the dimensions of the room are 5 ft x 12 ft.
To find the dimensions of the room, we can use the Pythagorean theorem since we have the length of the diagonal and one of the walls.
Let's assume that one wall measures x ft. Since the other wall is 7 ft longer, its length would be x + 7 ft.
According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides (walls in this case).
So, we can write the equation as:
x^2 + (x + 7)^2 = 13^2
Simplifying the equation, we get:
x^2 + (x^2 + 14x + 49) = 169
Combine like terms:
2x^2 + 14x + 49 = 169
Rearranging the equation and simplifying further:
2x^2 + 14x - 120 = 0
Now, we can solve this quadratic equation for x. We can either use factoring, completing the square, or the quadratic formula.
By factoring, we get:
2(x - 4)(x + 15) = 0
This equation will be satisfied when either (x - 4) = 0 or (x + 15) = 0.
So, we have two possible solutions:
1. x - 4 = 0
x = 4
2. x + 15 = 0
x = -15
Since dimensions cannot be negative, we discard the second solution.
Therefore, the length of one wall is 4 ft, and the length of the adjacent wall is 4 + 7 = 11 ft.
Hence, the dimensions of the room are 4 ft by 11 ft.
x^2 + (x+7)^2 = 13^2
So just solve for x and x+7
Or, save yourself some time and remember the 5-12-13 Pythagorean triple.