Draw a parallel through a given point

Draw a line running 12 squares across and 12 squares up, sloping at 45°. Mark a point two squares to the right of its lower end and six squares above that end. Construct the line parallel to the first that STARTS at that point — the point is its lower end — the same size and the same slope. Transfer the angle with arcs rather than judging it by eye.

What you are given

Each square is one unit. Draw these first, then construct from them.

The method

Worked example: the perpendicular bisector of a 6-unit line, with the two compass arcs left in. It illustrates the method only, not this exercise's answer.

Line

Geometric constructions: hints

Construction lines are drawn, not hidden
The arcs and rays you use to find a point are part of the drawing, not scaffolding to rub out. Draw them with the Construction line type: the marker ignores that type entirely, so they cost you nothing and they show your method.
Only some constructions land on a grid
This tool marks exact positions, so an exercise can only ask for a construction whose answer sits on whole grid points. Bisecting a right angle works, because 45° is a grid direction. Bisecting 45° does not, because 22.5° is not — and on this square grid a regular hexagon never does. That is why you will not find those here.
A bisector needs an even span
The midpoint of a line spanning an odd number of squares falls half-way between two grid points, where you cannot place it. Every line you are asked to bisect here spans an even number for that reason — count the squares before you look for the centre.
Divide half-width and height into the SAME number of parts
Split the base into equal divisions along the axis and the same number of divisions up the height. The method only produces a true parabola when both counts match — four divisions across needs four divisions up, not three or five.
Construction lines are working lines, not the answer
The grid of offset lines and the points they locate exist to find the curve — they are not part of the final drawing. What is graded is the curve itself, drawn as its own line through the located points, not the scaffolding that found them.
Number the divisions consistently on both sides
Label the height divisions 1, 2, 3... from the vertex outward, and use the same numbering on the base divisions. Point 2 on the height line only connects to point 2 on the base — mismatched numbering is the most common way this construction goes wrong.
On this sheet, join the points with straight segments, not a smooth sweep
On paper you would blend the located points into one smooth arc by eye. Here, draw a straight line from each point to the next, in order out from the vertex — that is what the marker compares your drawing against, point for point, so a hand-smoothed curve will not match it. This is a property of the tool, not of the geometry: more divisions make the joined segments read as a smoother curve on the sheet, even though every one of them is still a straight line underneath.

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