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CBSE Class 7 · Mathematics

Constructions and Tilings

Official NCERT chapter from Ganita Prakash Part II (book code gegp2). ExamMaster notes are original teaching at CBSE Class 7 depth.

This lesson follows the official chapter “Constructions and Tilings” in Ganita Prakash Part II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 7
  • Easy level
  • 8 concepts

1Geometric Constructions

A geometric construction is a sequence of allowed steps (ruler, compass, set-square as taught). The figure is the result of those steps, not a careful sketch.

A freehand triangle labelled “constructed” is the miss this chapter names later.

Figure. The finished SSS triangle ABC is the construction the chapter asks you to keep. Compass arcs are not drawn here. Tick marks on the three sides name the three given lengths; the figure is the meeting of those three segments.

How it works

  1. Name the allowed toolsThe rules of the game.
  2. Do the steps in orderArcs, copies, perpendiculars.
  3. The figure is what those steps producedA construction.

2Tiling

A tiling covers a patch without gaps or overlaps using copies of a shape (or a small set of shapes). The test is cover: can you continue the pattern and still fit?

A pretty border that leaves a hole is not a tiling of that patch.

Figure. A tiling covers a region with copies of one unit tile, edge to edge, with no gap and no overlap. The six identical rectangles are the same tile repeated. The covered rectangle is the join of those six copies.

How it works

  1. Choose a unit tileA shape.
  2. Copy it by the allowed isometries you were taught (shift, turn, flip)The cover.
  3. Check no gap and no overlap in the patchA tiling.

3A definition is a test you can run

Construction means you can rerun the steps and get the same figure (up to the same given). If you cannot list the steps, you have a drawing, not a construction.

“I drew it neatly” is not a step-list.

Figure. A definition is a test you can run. A tiling passes only if copies cover without a gap and without an overlap. The left panel passes. The right panel fails because a gap is left between tiles.

How it works

  1. Write the stepsThe definition’s object.
  2. Rerun themThe test.
  3. If a step is “sketch until it looks right”, it is not this constructionAllowed tools only.

4Name the given before the unknown

The given may be a line and a point, or a tile. The unknown is the constructed parallel, the perpendicular, or whether the tile works. Name the given marks first.

Assuming a 60° arc because equilateral was last chapter is a silent given.

Figure. Name the given before the unknown. Sides a and b are already marked. The unknown is angle C at the top. Construction starts from the two given lengths, not from a guess at C.

How it works

  1. Copy the given marksThe given.
  2. Name what to construct or decideThe unknown.
  3. Then run allowed stepsGiven first.

5One worked case is enough at this class

One constructed figure or one tile-check is enough. A page of the same perpendicular clone does not add a new step-list.

Ten twins of the same arc are still one construction.

Figure. One worked case at this class is enough: the finished perpendicular bisector of AB. M is the midpoint. The upright line through M meets AB at right angles. Compass arcs that located M are omitted.

How it works

  1. Run one construction (or one tile)The case.
  2. A second only as a checkOptional.
  3. StopThis class.

6A miss: swapping the school name for the picture

A tiled floor photo is a setting. The maths is whether your unit covers without gaps. Admiring the photo and skipping the gap-test is the swap-miss.

A mosaic picture is not a proof your pentagon tiles the plane.

Figure. The school name is not the picture. A wide rectangle can be called a square and still fail the square test: four equal sides and four right angles. The right panel is drawn as a visual square (width equals height over the 2.0 aspect) and passes.

How it works

  1. Keep the floor as a storyA setting.
  2. Name the unit and the patchThe maths.
  3. Check gaps and overlapsDo not swap.

7Check by the opposite action or the opposite test

Check a construction by a property: a perpendicular should pass a square-test; a copied segment should match on the compass. A check that is only “looks straight” is not the opposite test.

Staring at an arc and saying “round enough” is not a check.

Figure. Check a construction by the opposite test: the two segments you claimed equal must measure equal. Both brackets span 0.30 of the figure width, so AM and MB are the same drawn length.

How it works

  1. State the property you meant90°, or equal length.
  2. Test it with a toolThe opposite check.
  3. Mend a step if it failsHonest.

8Keep the claim at this chapter, not the next

This chapter constructs and tiles. It does not finish every later circle theorem. Keep the claim at allowed steps and gap-free cover.

Inventing a Class 9 circle theorem here is the next book stealing this one.

Figure. This chapter constructs and tiles. It does not yet prove similarity or chase angles in a later congruence lesson. Keep the claim at the finished figure and the tiling test.

How it works

  1. Stay with steps and tilesThis chapter.
  2. Get a figure or a yes/no coverThe job.
  3. Leave later theorems for their chapterNot the next.
A tiling of a patch means
  1. Copies cover it with no gaps and no overlaps
  2. A pretty border with holes
  3. A freehand sketch

Cover is the test.

Notes

  • The official chapter title is “Constructions and Tilings”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Recap

Hold these pegs from the official chapter “Constructions and Tilings”. The wording is ExamMaster’s teaching, not a textbook recap.

Geometric Constructions
A geometric construction is a sequence of allowed steps (ruler, compass, set-square as taught).
Tiling
A tiling covers a patch without gaps or overlaps using copies of a shape (or a small set of shapes).
A definition is a test you can run
Construction means you can rerun the steps and get the same figure (up to the same given).
Name the given before the unknown
The given may be a line and a point, or a tile.
One worked case is enough at this class
One constructed figure or one tile-check is enough.
A miss: swapping the school name for the picture
A tiled floor photo is a setting. The maths is whether your unit covers without gaps. Admiring the photo and skipping the gap-test is the swap-miss.

Practise Constructions and Tilings

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  • 1 quick check with worked explanations
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