The diagonal of a rectangular field
is 16 metres longer than its shorter side. If the longer side is 14 metres
longer than the shorter side, find the lengths of the sides of the field.
We should begin by sketching a
diagram of the field. Remember it doesn’t need to be to
scale. But it should be roughly in
proportion so we can identify how to solve this problem. It’s a rectangular field. So we know each of its interior
angles are 90 degrees.
We can also call the shortest side
of the field 𝑥 metres. The diagonal is 16 metres
longer. Let’s call that 𝑥 plus 16
metres. Since the longer side is 14 metres
longer than the shorter side, we can call that 𝑥 plus 14 metres. Notice that by including the
diagonal of the rectangular field, we’ve created two identical right-angled
triangles. This means we can use the
Pythagorean theorem to help us calculate the value of 𝑥.
Remember the Pythagorean theorem
says that the square of the hypotenuse — that’s the longest side — is equal to the
sum of the squares of the two smaller sides. It’s often written as 𝑎 squared
plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the length of the hypotenuse.
Let’s substitute what we know about
our right-angled triangle into this formula. The diagonal is the hypotenuse;
that’s 𝑥 plus 16. So our formula becomes 𝑥 squared
plus 𝑥 plus 14 squared equals 𝑥 plus 16 squared.
Our next step is to expand the
brackets before we do anything else. A common mistake here is to think
that to square a bracket, we just square its two terms. In fact, we should write it as the
product of two brackets and then expand fully.
We first multiply the first term in
each bracket. 𝑥 multiplied by 𝑥 is 𝑥
squared. We then multiply the outer
terms. That’s 𝑥 multiplied by 14, which
is 14𝑥. Multiplying the inner terms —
that’s 14 multiplied by 𝑥 — is once again 14𝑥. And then, 14 multiplied by 14 is
196. It’s also sensible to simplify by
collecting like terms. That gives us 𝑥 squared plus 28𝑥
We can repeat this process for 𝑥
plus 16 squared. 𝑥 multiplied by 𝑥 is 𝑥
squared. 𝑥 multiplied by 16 is 16𝑥. 16 multiplied by 𝑥 gives us
another 16𝑥. And 16 multiplied by 16 is 256. This simplifies to 𝑥 squared plus
32𝑥 plus 256. Substituting these back into our
original equation and we get 𝑥 squared plus 𝑥 squared plus 28𝑥 plus 196 is equal
to 𝑥 squared plus 32𝑥 plus 256.
We have a quadratic equation. And to solve this, we first need to
rearrange so that we have a quadratic expression that’s equal to zero. Let’s subtract 𝑥 squared from both
sides. Doing so leaves us with 𝑥 squared
plus 28𝑥 plus 196 is equal to 32𝑥 plus 256. Next, we’ll subtract 32𝑥 from both
sides. And that leaves us with 𝑥 squared
minus four 𝑥 plus 196 equals 256. Finally, we’ll subtract 256 from
both sides. That gives us 𝑥 squared minus four
𝑥 minus 60 is equal to zero.
Now, we need to factorize the
expression on the left-hand side. We know that it will consist of two
brackets. And in the front of each bracket,
there will be an 𝑥. We know this because when we
multiply these brackets back out again, 𝑥 multiplied by 𝑥 gives us this 𝑥 squared
To find the constant in each
bracket, we need to find the factor pair of negative 60 that has a sum of negative
four. Negative 60 has quite a few factor
pairs. Let’s narrow it down to 10 and
negative six or six and negative 10. We’ve done this because 10 and six
have a difference of four. In fact, the factor pairs that sums
to make negative four is six and negative 10.
So factorizing our expression, we
get 𝑥 plus six multiplied by 𝑥 minus 10 is equal to zero. Now, we have two brackets whose
product is zero. For this to be true, either bracket
must individually be equal to zero: so 𝑥 plus six is equal to zero or 𝑥 minus 10
is equal to zero.
To solve this first equation, we’ll
subtract six from both sides. And that gives us 𝑥 is equal to
negative six. To solve the second equation, we’ll
add 10 to both sides. That gives us that 𝑥 is equal to
10. In fact, we said that 𝑥 was the
length of the shorter side of the field. 𝑥 cannot, therefore, be
That means that the shorter side of
the field must be 10 metres in length. The longer side of the field is 14
metres longer. 10 plus 14 is 24. So the longer side of the field is