i know that i use the pythagorean theorem x^2 + (3x-1)^2 = 37 but i don't know how to solve that
The diagonal of a rectangle is 37. The length is 1 less than the width. How do I find the perimeter?
- Posted:
- 3+ months ago by anonymous...
- Topics:
- geometry, homework, perimeter, diagonal, math, mathematics, length, pythagorean theorem
Details:
Answers (2)
Let the width be (x) units... Therefore, the length will be (3x-1) units
Diagonal^2 = (Side1)^2 + (Side2)^2
therefore, (37)^2 = (x)^2 + (3x-1)^2
therefore, 1369 = x^2 + 9x^2 - 6x + 1
therefore, 1369 = 10x^2 - 6x + 1
therefore, 10x^2 - 6x - 1368 = 0
therefore, 5x^2 -3 x - 684 = 0 ... dividing by 2
therefore, 5x^2 - 60x + 57x - 684 = 0 ... splitting the middle term
therefore, 5x(x - 12) + 57(x - 12) = 0
therefore, (x - 12) (5x + 57) = 0
Therefore, either x - 12 = 0 OR, 5x + 57 = 0 ... If ab=0 then either a=0 or b=0
Therefore, x = 12 OR x = -57/5
But Length cant be negative... Therefore, x = -57/5 is discarded
therefore x = 12
so, width is 12 and length is (x - 1) = 11
perimeter of rectangle = 2(length + width) ... formula
therefore, perimeter = 2(12 + 11)
therefore, perimeter = 2 * 23
therefore, perimeter = 46 uints
Plz comment of this
At first I was a bit puzzled when I saw this peculiar (3x - 1)². Well, the solution, a:= width = 12 and hence, as b equals a -1, b:= length = 11, is wrong. Go test it:
sqrt(11² + 12²) = 37??? Nope, is just 16.2788something. So, the condition is not fulfilled.
As the length, b, equals width a - 1, we can express b in terms of a, which we shall do henceforth throughout this calculation. Get yourself a cup of tea and lo and behold:
Pythagoras tells us that a² + (a-1)² = 37²
(a-1)² is just (a-1)(a-1) = a² - 2a + 1. Together with the width² this sums up to:
2a² - 2a + 1 = 37² = 1369
2a² - 2a = 1368 | :2
a² - a = 684
a² - a - 684 = 0
That's a simple quadratic equation, which can be solved. We get two solutions
one positive and one negative. We ignore the negative, as negative lengths of rectangles don't make much sense.
a = 26.65817281...
b = a - 1 = 25.65817......
test it! Now the diameter of the rectangle has the demanded length of 37.
sqrt = square root
sqrt(a² + b²) = 37
That's it.