The points one, zero and four, four
are two vertices of a rhombus. One of the edges of the rhombus
lies along the positive 𝑥-axis. Work out the coordinates of the
other two vertices of the rhombus.
The vertices of any shape are the
points or corners. In this question, we’re told that
two of them have coordinates one, zero and four, four. The two vertices must be joined by
an edge or side of the rhombus.
A rhombus is a four-sided shape
with the following properties. All four sides are equal in
length. It has two pairs of parallel
sides. And the opposite angles are equal,
none of which are equal to 90 degrees.
Let’s consider the first property
that all four sides are equal. In order to get from the point one,
zero to four, four, we need to go four units up our grid and three units to the
right. This creates a right-angled
To calculate the missing length of
any right-angled triangle, in this case labelled 𝑥, we can use Pythagoras’s
theorem. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is the longest length of the triangle,
called the hypotenuse.
The hypotenuse of our triangle is
𝑥. Therefore, 𝑥 squared is equal to
three squared plus four squared. Three squared is equal to nine, as
three multiplied by three is nine. And four squared is equal to
16. Adding nine and 16 gives us 25. Therefore, 𝑥 squared is equal to
25. The opposite or the inverse
operation of squaring is square rooting. Therefore, we need to square-root
both sides of this equation to work out 𝑥.
The square root of 25 can be
positive or negative five. As we’re dealing with a length, 𝑥
must be equal to positive five. The length of the line joining the
coordinates one, zero and four, four is five units.
We were told in the question that
one of the edges of the rhombus lies along the positive 𝑥-axis. This means that we need to go five
units to the right from the point one, zero. One plus five equals six. This means that one of the other
vertices of the rhombus is six, zero.
One of the other properties of a
rhombus was that opposite sides are parallel. Therefore, one of the sides must be
parallel to this line. There must be a horizontal edge
five units long from four, four. Four plus five is equal to
nine. Therefore, the fourth vertex must
have coordinates nine, four.
The final edge of our rhombus must
join the points six, zero and nine, four. This edge will be parallel to the
line from one, zero to four, four. Once again, it will be of length
five units. The four vertices of the rhombus
have coordinates one, zero; four, four; six, zero; and nine, four. The two that we were not given in
the question are six, zero and nine, four.