The perimeter of a rectangle
is 70 centimeters, its
diagonal is 25 centimeters.
Find the length and the width.
So if this is our rectangle,
this is the length
and this is the width,
and the opposite sides
have the same length, so
this is also the length
and this is also the width.
We know the diagonal of this link here
is 25 centimeters, and we also know
the perimeter or the distance around
the rectangle is 70 centimeters.
Because the perimeter is 70 centimeters,
we know that 2L plus 2W equals 70.
Now from here, let's write the length L
in terms of W by solving
this equation for L.
The first step is to isolate the L term
by subtracting 2W on both sides.
Simplifying, we have 2L minus 2W is zero,
so we have 2L equals 70 minus 2W,
and now we divide both sides by two,
or divide each term by two.
Simplifying here, we have 1L or L equals,
70 divided by two is 35.
And here, we have two divided by two,
which simplifies the one,
giving us just minus W.
So now, we can say that
L is equal to 35 minus W.
Remember our goal here is to determine
the length and the
width of this rectangle.
Well notice the diagonal
cuts the rectangle
into two right triangles.
Let's focus on this
upper-right triangle here.
We can set up an equation
in terms of W now using
the fact that for our right triangle,
A squared plus B squared equals C squared,
where A and B are the lengths of the legs
and C is the length of the hypotenuse.
So notice how, for the
Pythagorean Theorem,
this is the hypotenuse, C, and these
would be the legs of the right triangle,
let's call this A, and let's call this B.
Which means W squared plus,
again we want one equation
in terms of one variable,
so we won't use L,
we will use 35 minus W.
So we have plus the square of 35 minus W
must equal 25 squared.
Now we will simplify and solve for W.
We have W squared plus
two factors of 35 minus W,
equals 25 squared, which is 625.
Now, to multiply the two binomials,
where we have four products,
one, two, three, and four.
We have W squared plus
35 times 35 is 1,225.
And then we have 35 times negative W,
which is negative 35W,
and then this product
also gives us a negative 35W.
So we have minus 70W, and then negative W
times negative W is W squared, so we have
plus W squared equals 625.
Simplifying the left side of the equation,
we have two like terms here,
which gives us 2W squared
minus 70W, and then we
have plus 1,225 equals 625.
And now we subtract 625 on both sides,
which gives us 2W squared minus 70W.
1,225 minus 625 is 600, so
we have plus 600 equals zero.
Notice how we have a quadratic equation,
which we can now solve by factoring.
Remember, the first step in factoring,
is to factor of the
greatest common factor,
which in this case is two.
If we factor out two,
we're left with W squared
minus 35W plus 300.
If this quadratic in the
parentheses does factor,
it will factor into two binomial factors.
Because we have a W squared here,
which is equal to W times W.
We have a factor of W here
and a factor of W here.
And the constant terms,
or the second terms
in each binomial will
be the factors of 300
that add to negative 35.
That's not an easy
question, but negative 15
times negative 20 is positive 300.
And negative 15 plus
negative 20 is negative 35,
which means one binomial factor
is W minus 15, and the
other binomial factor
is W minus 20.
Now, using the zero product property,
if this product is equal to zero,
either W minus 15 equals zero,
or W minus 20 is equal to zero.
Solving for W, we have W equals 15,
or W equals 20.
Now we have the information we need
to answer the question.
The length, L, again
is equal to 35 minus W.
Let's use W equals 15,
which gives us 35 minus 15,
which equals 20, and this is the length.
It would be 20 centimeters.
And the width, which is equal to W, is 15,
and of course, this is centimeters.
And I want to show what
happens if we use W equals 20.
If we use W equals 20, we would have
the length is equal to 35 minus 20,
which is equal to 15 centimeters.
And the width, if we're using W
equals 20, is 20 centimeters.
But typically, if the
length is a longer side,
and the width is a shorter side,
and therefore, we will say the length
is 20 centimeters, and the
width is 15 centimeters.
I hope you found this helpful.
