Week 01 · Points, Lines and the Chalk Grid

Tuesday

Seven constructions

// Precise words and string-and-chalk constructions

⏱ about 20 min

Tuesday: Seven Constructions

Wren pins a page to the corkboard: seven small drawings, each made of arcs and straight lines. "Our construction kit."

"Copy a segment, copy an angle, bisect both, a perpendicular, a parallel, and shapes in a circle," Comet reads. "What can we make?"

"The back wall line first," Wren says. "Then a perpendicular through the stage mark, so the first flat stands square."

Comet swings two arcs from the mark along the wall line. Then two bigger arcs from those crossings. They meet above.

Nova projects a right-angle mark where the new line meets the wall. "Would you like a hint? Check it with the carpenter's square."

"Square," Comet says, grinning. "The compass never lies."

"It never guesses," Wren says. "Our designer, every step has a reason. Find them."

The construction kit

A construction uses only two tools: a compass (our string and chalk) to draw arcs, and a straightedge to draw lines.

The compass copies a distance. The straightedge joins two points. No measuring, no protractor, no guessing.

Construction	Key move	Why it works
copy a segment	set the compass to the segment, mark that distance on a new line	the compass keeps one distance
copy an angle	one arc across both rays, then copy the arc and its width	equal arcs and equal chords give an equal angle
bisect a segment	equal arcs from both ends cross above and below	each crossing is equally far from both ends
bisect an angle	one arc across both rays, then equal arcs from the two crossings	the new crossing is equally far from both rays
perpendicular at a point	equal arcs either side of the point, then arcs from those crossings	it is the perpendicular bisector of the two side points
parallel through a point	draw a slanted line through the point, then copy its angle at the point	equal corresponding angles mean parallel lines
hexagon in a circle	keep the compass at the radius and step six times around	each step is one side equal to the radius

$An angle at V with one arc across both sides, two more arcs crossing inside, and the dashed bisector.$

Bisect an angle, step by step

Set the compass at the vertex V and swing one arc that crosses both rays. Call the crossings S and T.
Keep the same radius. Swing an arc from S and an arc from T so they cross inside the angle at X.
Draw the ray from V through X with the straightedge. That ray is the bisector.
Check: the angle on each side of the bisector is half the original.

$A line with point P, equal arcs either side of P, two arcs crossing above, and the dashed perpendicular at P.$

$A line, a point P above it, a slanted line through P, two copied angle arcs and the dashed parallel through P.$

Why it works, and two methods for one job

Why the angle bisector works: S and T are the same distance from V. X is the same distance from S and from T.

So the two halves match side for side, and matching triangles have matching angles at V.

A parallel line through P can be built two ways. One: copy an angle so the corresponding angles match.

Two: build a perpendicular to the line, then a perpendicular to that perpendicular through P.

Both give the same line. The angle copy uses fewer arcs. The double perpendicular is easier to check with a square.

BISECT IT

Read the question.
Tap your answer.

The angle between two chalk lines measures 70°. Comet bisects it with string and chalk. How many degrees is each half?

Wren bisects the 120° angle between two turntable spokes. How many degrees is each half?

A chalk line crosses the wall line and makes a 70° angle on one side. What is the angle on the other side?

WHICH CONSTRUCTION?

Read the question.
Tap your answer.

$A line with point P, equal arcs either side of P, two arcs crossing above, and the dashed perpendicular at P$ Comet needs a flat to stand at a right angle to the wall line through one mark. Which construction does she use?

$A line, a point P above it, a slanted line through P, two copied angle arcs and the dashed parallel line through P$ Wren wants a second chalk line that never meets the wall line and passes through a stage mark. Which construction?

$Segment AB with two equal compass arcs from A and B crossing above and below, and the perpendicular bisector through the midpoint M$ The crew wants the exact middle of a chalk segment without a tape measure. Which construction?

BISECT AN ANGLE IN ORDER

?Swing one arc from the vertex that crosses both rays at S and T
?Swing equal arcs from S and from T so they cross inside the angle at X
?Draw the ray from the vertex through X
?Check that the two new angles are equal

WHY THIS EXERCISEThe order matters because each arc uses points made by the arc before it.

Two lines that meet at a right angle are called this. Type the one word.

To split a segment or an angle into two equal parts is to do this. Type the one word.

Try it

On paper, draw an angle. Bisect it with a pencil on a string, following the four steps.
Check both halves with a protractor. Write how close they are.

Draw a line with a point P on it. Construct the perpendicular at P with arcs only, then label every arc.

Shop Book: write the four steps for bisecting an angle, and beside each step write its reason.Pick one other construction from the table. Draw it small and label the arcs in the order you would draw them.The parallel line can be built two ways. Write which way you would choose in the shop, and why.

Excellent work with the kit. Tomorrow is Shop Lab: string, chalk and a circle with six marks.

Grown-up checked my answers

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