PrepShorts · Teaching notes · Class 7 Mathematics · Chapter 6, Constructions and TilingsPrepShorts

Chapter 6 · Constructions and Tilings

Constructing a 90° angle at a chosen point on a line

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9 min.

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What to assume they know

What they should be able to do

  • State the problem: a right angle wanted at a named point of a given line
  • Explain why extending the line and marking two equal distances from O turns the new problem into the old one
  • Say why the perpendicular bisector of the manufactured segment is bound to pass through O
  • Explain why a single crossing pair of arcs now suffices
  • Carry out the construction with a compass and an unmarked ruler
  • Identify what the second arc pair was doing in the earlier construction, and why it can be dropped here
  • Adapt the same reversal to drop a perpendicular from a point lying off the line
  • Recognise the 90° angle as the seed that later gives 45°

Where it usually goes wrong

  • "This is a fresh construction to be learnt separately." It is the earlier one, read from the other end. If the explanation presents it as new, the student will carry two recipes instead of one idea.
  • **"You must draw arcs above and below."** Here you do not need to. The chapter asks the question explicitly on p.141 and answers no, because O is already a known point of the line being built.
  • "X and Y have to be at some particular distance from O." Any opening will do; only equality matters. Students who think a special length is required will reach for a scale, which the chapter has just argued against.
  • "The right angle is at A." It is at O. A is only the second point that fixes the line's direction.
  • "A perpendicular through a point off the line needs a different method again." Question 4 on p.155 is the same reversal a third time: find a segment of the line that P is equally far from both ends of.
  • "Right angles come from a set square or a protractor." The chapter has restricted itself to two tools from the foot of p.140 onward, and everything after that point respects the restriction.

Questions to check understanding

  • Construct a right angle at a marked point of a given line, using only a compass and an unmarked ruler
  • Explain why the construction begins by extending the line
  • State how many crossing arc pairs this construction needs, and justify the number
  • Given a line and a point not on it, construct the perpendicular through that point (Figure it Out question 4, Part II, p.155)
  • Describe a rope procedure for a right angle at a point (Figure it Out question 2, Part II, p.142)
  • The 90°-angle bullet of the SUMMARY (Part II, p.163) is the statement the chapter expects back
  • Constructions that begin from a right angle — 45°, and the eight-armed design — are examined in Bisecting an angle, and halving 90° to get 45°

Examples worth working on the board

  • The opening figure, and Fig. 6.2 (Part II, §6.1, p.141). Checked against the printed page. Two drawings sit one above the other, but only the lower one carries the caption. The upper drawing — a bare horizontal line with a single dot on it lettered O — is uncaptioned and belongs to the opening question. Fig. 6.2 is the drawing below it: the same line with the dot still lettered O and two further points marked, X to its left and Y to its right, each carried by a small arc bracket that shows they were stepped off with the compass from O.
  • Fig. 6.3 (Part II, §6.1, p.141). Checked against the printed page. The same line with X, O, Y on it; a single pair of arcs crossing above the line at a point lettered A; the segment from A down to O drawn in; and a small square symbol at O marking the right angle. There is no crossing below the line — that is the visual payoff of the whole topic and the figure must not add one.
  • The counting argument (Part II, §6.1, p.141). Inputs: a line needs two points; O is already known to be on the wanted line; one pair of arcs supplies one more.
  • Setting the compass. Any opening works, provided it is used unchanged for X and for Y — that is what makes O the midpoint. The chapter does not print a length here; show the compass locked.
  • The outside-point version (Part II, §6.1, "Figure it Out", question 4, p.155). Checked against the printed page: the question gives a line l and a point P marked away from it, and its printed hint tells the reader to look for a segment of l whose perpendicular bisector runs through P.
  • The rope version (Part II, §6.1, "Figure it Out", question 2, p.142). The chapter asks, under its Math Talk flag, for methods of building a right angle at a point using a rope, and prints no method. Treat it as an open prompt.
  • The 90° that is about to be halved (Part II, §6.1, p.143). The next subheading needs 45°, and gets it by bisecting the angle built here. Say so at the end; it is why this construction matters.

Figures to have open

  • A horizontal line with a movable point O, able to be shown moving so the compass can step off X and Y symmetrically. Standard schematic; this is the topic's key image.
  • A side-by-side of the four-arc and two-arc constructions, built fresh — the book prints the two constructions one page apart (pp.139–140 and p.141) and never sets them together. The substantive point is the never-together, not the distance.
  • The outside-point construction: a line l, a point P above it, an arc from P cutting l at two points, and the bisector of that cut segment passing through P. Standard schematic.
  • No photograph, table or dataset from the textbook is needed for this topic.

Where this sits in the book

The book

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