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Lesson 16 of 17

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9 min

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Thirteen quick questions. Use the table if you get stuck.

Thirteen checks for a construction
IdeaThe check
Copy a cornerSame radius. Carry the gap.
The perpendicular bisectorEvery point is the same distance from both ends.
Split a cornerBoth marks must be at the same distance.
A parallelMeasure square across at two places.
Three lengthsEach length must be less than the other two added.
Two lengths, angle betweenThe angle must sit between the two lengths.
A tilingNo gaps and no overlaps.
The whole turnTile corners at one point add to 360°.
A square lineTo draw a perpendicular, you need a segment on the line. Its perpendicular bisector must go through your point. On the line, make the point the middle of the segment. Off the line, pick ends that are equally far from it.
60°, 30° and 15°Two arcs of one span make an equilateral triangle. Its corner is 60°. Bisect a corner to halve it. Bisect 60° to get 30°. Bisect 30° to get 15°.
A hexagonDraw a circle. Keep the compass at the radius. Step it round the circle. Six steps come back to the start. Join the six points. That is a regular hexagon.
The tangramA tangram is a square cut into seven pieces. Two big triangles are 1/4 each. The medium triangle is 1/8. So are the little square and the slanted piece. Two small triangles are 1/16 each. The shares add to 1.
Tiling the whole floorUse copies of one regular shape, edge to edge. They tile when whole corners fill 360°. Divide 360° by the corner. A whole number means they fit. Only the triangle, square and hexagon do.

You carry the gap correctly. But the new arc has a bigger radius. Is the copy the same corner?

  • Yes. The gap is the same size
  • No. The radius must stay the same
  • Yes, if the corner is a right angle

A point is 6 cm from A and 6 cm from B. Where does it lie?

  • At the middle of AB, and nowhere else
  • On the line through A and B
  • On the perpendicular bisector of AB

Marks are 25 mm and 35 mm out on the two arms. You finish the construction. What do you get?

  • A ray, but two unequal halves
  • Two equal halves, as usual
  • No ray at all, since the marks differ

Two lines do not cross anywhere on your page. Are they parallel?

  • Yes. They never meet on the page
  • Not sure. Measure across at two places
  • No. Parallel lines must cross somewhere

A shop has strips of 3 cm, 4 cm and 9 cm. Do the arcs meet?

  • Yes. They make a thin triangle
  • Yes, if the 9 cm strip is the base
  • No. 3 + 4 is less than 9

A 40° corner is at the end of a 60 mm side. A 45 mm side sits across from it. How many triangles fit?

  • Exactly one
  • Two
  • None

Can 2 × 1 tiles cover a 5 × 5 grid?

  • No. 25 squares is odd
  • Yes. Turn the tiles
  • Yes, with one gap allowed

Three corners of 108° meet at a point. Do they fill it?

  • Yes. Three corners are enough
  • No. They add to more than 360°
  • No. 324° is 36° short

A point is 40 mm above a line. You strike an arc with the compass at 50 mm. How does the arc meet the line?

  • It cuts the line twice
  • It misses the line
  • It touches the line once

You start with 60°. You bisect it, then add 60°. What corner do you get?

  • 120°
  • 45°
  • 90°

A circle has a radius of 45 mm. You step a 45 mm span round it. How long is each side of the hexagon?

  • 22.5 mm
  • 45 mm
  • 90 mm

Which pieces together make exactly half of the tangram square?

  • The medium triangle, the little square and the slanted piece
  • The two small triangles and the little square
  • The two big triangles

Which regular shape, used alone, can never fill a point exactly?

  • The shape with 8 sides
  • The triangle
  • The hexagon

Chapter 6 of Part II is Constructions and Tilings. Try its designs with a compass.

Next: Finding the Unknown. What an equals sign claims. How to work back to a hidden number.

Reveal every answer to finish this lesson.