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This investigation looks at the distances from the origin (0, 0) to the centres of squares and hexagons that form a pattern.
The diagram shows a pattern of congruent squares.
A dot marks the centre of each square.
The coordinates of some of the centres are (0, 0), (-3, -2) and (3, 1).
(a) Complete this table, using a tick in each row, to show whether a point is
• at the centre of a square
• inside a square, but not at its centre
• on the side of two squares
• where four squares meet.
The first three have been done for you.
[Table_1]
$$\begin{array}{|c|c|c|c|c|}\hline \text{Point} & \text{At the centre of a square} & \text{Inside a square, not at its centre} & \text{On the side of two squares} & \text{Where four squares meet} \\ \hline (0, 0) & \checkmark & & & \\ \hline (-1, 0.5) & & & \checkmark & \\ \hline (0.5, 0.5) & & \checkmark & & \\ \hline (1.5, 1.5) & & & & \\ \hline (0, 1.5) & & & & \\ \hline (0.25, 0) & & & & \\ \hline (100.5, 99.5) & & & & \\ \hline \end{array}$$
[4]
(b) Give a reason for your answer for (100.5, 99.5).
.............................................................................................................................................................................
............................................................................................................................................................................. [1]
533
507
578
Each dot on the grid marks the centre of a square.
The nearest dots to $(0, 0)$ are $(-1, 0)$, $(0, 1)$, $(1, 0)$ and $(0, -1)$.
These four dots are the 1st nearest neighbours.
The 2nd nearest neighbours to $(0, 0)$ are $(1, 1)$, $(1, -1)$, $(-1, -1)$ and $(-1, 1)$.
All nearest neighbours have integer coordinates.
(a) One of the 3rd nearest neighbours to $(0, 0)$ is $(2, 0)$.
Find the other 3rd nearest neighbours and write down their coordinates.
......................................................................................................................... [3]
(b) Find the coordinates of all the 4th nearest neighbours to $(0, 0)$.
..........................................................................................................................
.......................................................................................................................... [3]
571
584
517
You can find the distance, $d$, from $(0, 0)$ to the point $(a, b)$ using Pythagoras’ Theorem.
$$d^2 = a^2 + b^2$$
(a) Show that the distance of a 4th nearest neighbour from $(0, 0)$ is $\sqrt{5}$. [2]
(b) Here are four points and their coordinates.
\( A(20, 5) \quad B(7, 24) \quad C(-7, 24) \quad D(0, 25) \)
Which of these points are a distance of 25 units from $(0, 0)$? [3]
(c) There are more than 10 nearest neighbours to $(0, 0)$ with $d = 5$.
Four of them are $(0, 5)$, $(5, 0)$, $(-5, 0)$ and $(0, -5)$.
On the grid, mark with a cross all the nearest neighbours to $(0, 0)$ with $d = 5$.
[4]
518
500
533
(a) (i) Explain why the triangle shown is an equilateral triangle.
............................................................................................................................. [1]
The dots at the centre of the hexagons have coordinates as shown on the grid below.
(ii) A line is made by joining the points (3, 0) and (4, 1).
Work out the size of the acute angle between this line and the x-axis.
................................................... [2]
(b) The point at the centre of each hexagon has six 1st nearest neighbours.
(i) Complete this list of the 1st nearest neighbours to the point (1, 1).
(0, 1), (0, 2), (1, 2), ........................................................... [2]
(ii) Find, in terms of $a$ and $b$, the coordinates of the six 1st nearest neighbours to the point ($a$, $b$).
........................................................................................
........................................................................................ [3]
(c) A student suggests that an estimate of $d$, the distance from (0, 0) to the point ($a$, $b$), is $d = \sqrt{a^2 + b^2 + ab}$.
The distance between any point and its 1st nearest neighbours on this grid is 1 cm.
Does the student’s formula give a good estimate for the distance from (0, 0) to (4, -3)? Show how you decide.
529
547
545
The circles show the 1st, 2nd, 3rd, and 4th nearest neighbours to the point X.
(a) Complete this table.
[Table_1]
\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Nearest neighbour} & 1\text{st} & 2\text{nd} & 3\text{rd} & 4\text{th} & 5\text{th} & 6\text{th} & 7\text{th} & 8\text{th} \\ \hline \text{Number of nearest neighbours} & & & & & 6 & 12 & 6 & \\ \hline \end{array}
(b) A computer calculates that there is a total of 9 points for a certain nearest neighbour distance.
Explain why the computer is probably wrong.
585
552
559
