Video: AQA GCSE Mathematics Foundation Tier Pack 2 • Paper 2 • Question 9

The points (1, 0) and (4, 4) 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.

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Video Transcript

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 triangle.

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.

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