The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 4 cm, what are the dimensions (the length and the width) of the rectangle?
using 2 equations 2 unknown technique: 2l + 2w =27 -eq a million l x w =35 -eq 2 using eq2 . l x w = 35 l = 35/w -eq 3 use eq 3 in eq a million 2l + 2w = 27 2( 35/w ) +2w =27 70/w +2w =27 (2w)sqrd + 27w - 70 = 0 Use quadratic equation to remedy for w. once you get w, remedy for l using eq 3
Comments
Hi
Let the width = x, then
Length = x + 1, from the Pythagoras theorem it follows that
(Diagonal of rectangle)^2 = (its length)^2 + (its width)^2
4^2 = (x +1)^2 + x^2 .............|Diagonal. = 4cms
x^2 + 1 + 2x + x^2 = 16
2x^2 + 2x - 15 = 0
This is a quadratic equation of the form ax^2 + bx + c = 0,
whose Discrimnant D = b^2 - 4ac = 4 -4(2)(-15) = 124
As D is not a perfect square the eqn. cannot be solved
by factorization.
Using formula for quadratic equation,
x = [- b +/- sqrt(D)]/2a
x = [-2 +/- sqrt(124)]/4
x = [-2 +/- sqrt(4*31)]/4 = -(1/2) +/- (1/2)sqrt(31)
as the length of diagonal can not be a negative value
x = -(1/2) + (1/2)sqrt(31)
So width of rectangle = (1/2)[sqrt (31) - 1] = 2.28 cm
Length of rectangle = x + 1 = (1/2)[sqrt (31) + 1] = 3.28cm
Good luck
For the person with the "retarded" comment .... This is a pretty hard little question actually!
The answer above of 1.5 and 2.5 is wrong... sorry. 1.5 + 2.5 equals 4 ... that is the sides just stretch the length of the diagonal
Pythagorus theorem sends you around in circles. So trigonometry would be an easier way of solving it.
Take the corner angle and call it a. We'll call the adjacent Length and the opposite width and W=L-1
We know cos(a) = L/4
sin(a) = (L-1)/4
Then L = 4sin(a)-1
and L=4cos(a)
4cos(a)=4sin(a)-1
4sin(a)-4cos(a)=1
sin(a)-cos(a) = 0.25
I don't have trig tables with me but find that angle with them and then tan(a) will give you the proportions of the sides to one another.
width = 1.5, length =2.5
use this formula
A squared+ B squared = C squared.
the diagonal is the hypotenuse ( C squared)
width = X
length = X + 1
plug it all in.
using 2 equations 2 unknown technique: 2l + 2w =27 -eq a million l x w =35 -eq 2 using eq2 . l x w = 35 l = 35/w -eq 3 use eq 3 in eq a million 2l + 2w = 27 2( 35/w ) +2w =27 70/w +2w =27 (2w)sqrd + 27w - 70 = 0 Use quadratic equation to remedy for w. once you get w, remedy for l using eq 3
16=(l)squared + (l+i)squared
are you retarded?