Summary, Formula Sheet & Quick Quiz പുനരവലോകനം
All twelve key results of the chapter with pictures, the construction recipes, the proof tools, the three centres of a triangle, the mistakes to avoid, flashcards, and a scored quiz to find out what needs another look.
1. The key results
Equal division: equally spaced parallels cut every line into equal parts; parallels that cut one line equally are equally spaced, so they cut every line equally.
Ratio division: three or more parallels cut any two lines in the same ratio: a/b = p/q. Proof by areas of triangles.
Dividing a triangle: a line parallel to one side cuts the other two in the same ratio (the same fractions of themselves); and a line that cuts two sides in the same ratio is parallel to the third.
Midpoint theorem: the line through the midpoint of one side parallel to another bisects the third; the line joining two midpoints is parallel to the third side and half of it. Four congruent triangles.
Altitudes meet at the orthocentre (ലംബകേന്ദ്രം): they are the perpendicular bisectors of the bigger triangle made by parallels through the vertices.
Medians meet at the centroid (മധ്യമകേന്ദ്രം), which divides each median 2 : 1 from the vertex (parallelogram PQDE proof).
2. Construction recipes
- Divide AB into n equal parts: ray from A; n equal compass steps A₁ … Aₙ; join AₙB; parallels through the other marks (set squares). Each part = AB ÷ n.
- Divide AB in the ratio m : n: same ray with m + n equal steps; join the last mark to B; one parallel through mark m. Parts = AB·m/(m + n) and AB·n/(m + n).
- Triangle / square / hexagon of perimeter p: divide a line of length p into 3 / 4 / 6 equal parts; build the figure on one part (arcs for the triangle; perpendiculars for the square; a circle of radius = side for the hexagon).
- Rectangle with perimeter p and sides m : n: divide a line of length p/2 in m : n; the two parts are the sides.
- Triangle with perimeter p and sides a : b : c: divide a line of length p in a : b : c; swing arcs of the other two parts from the ends of one part.
- 7 cm into 3 equal parts (textbook): four parallels 2 cm apart; a 7 cm compass arc from a point on the first cuts the fourth; join. Each part 2⅓ cm.
3. Proof toolbox
Congruent right triangles
Between parallels, the little right triangles on a transversal have equal angles (corresponding) and either equal vertical sides (equal spacing) or equal hypotenuses (equal pieces). ASA / AAS makes them congruent – the engine of Keys 1–3.
Areas with the same height
Triangles with the same apex and bases on one line have areas in the ratio of their bases; triangles on the same base with apexes on a parallel have equal areas. Together they prove a/b = p/q (Key 4).
The ratio theorem twice
Most Exercise 3 proofs apply "a parallel cuts the other two sides in the same ratio" in two triangles and compare: AP/PB = AQ/QC and AQ/QC = AR/RD ⇒ AP/PB = AR/RD.
One pair equal and parallel
A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram; its diagonals bisect each other. Used for the medians (PQDE) and for Varignon's parallelogram.
4. The three centres
| Centre | Made by | Facts to remember |
|---|---|---|
| Orthocentre ലംബകേന്ദ്രം | the three altitudes (ഉന്നതികൾ) | inside an acute triangle, at the right-angle vertex of a right triangle, outside an obtuse one; the orthocentre of ΔHBC is A. |
| Centroid മധ്യമകേന്ദ്രം | the three medians (നടുവരകൾ) | divides each median 2 : 1 from the vertex; balance point; the six small triangles have equal areas; the medial triangle has the same centroid. |
| Circumcentre പരിവൃത്തകേന്ദ്രം | the perpendicular bisectors of the sides | equidistant from the vertices; for a right triangle it is the midpoint of the hypotenuse (circumradius = half the hypotenuse). |
| Euler line: circumcentre, centroid and orthocentre are collinear; the centroid divides the segment from the circumcentre to the orthocentre in 1 : 2. In an equilateral triangle all three coincide. | ||
5. Formula sheet
| Situation | Fact | Where |
|---|---|---|
| Equally spaced parallels | every transversal cut into equal parts | p. 5 |
| Any parallels (3 or more) | a/b = p/q on any two transversals | p. 6 |
| Line ∥ a side of a triangle | cuts the other sides in the same ratio and the same fractions: p/a = r/b, q/a = s/b | p. 8 |
| Converse | same ratio on two sides ⇒ parallel to the third | p. 8 |
| Midline | parallel to the third side, length = half; medial triangle: perimeter ½, area ¼ | p. 8, p. 9 |
| Midpoints of a quadrilateral | a parallelogram with sides ½ the diagonals; rhombus if diagonals equal, rectangle if perpendicular, square if both | p. 9 |
| Centroid | AG = ⅔ AD, GD = ⅓ AD | p. 10 |
| Right triangle | circumcentre = midpoint of the hypotenuse; distance to each vertex = ½ hypotenuse | p. 11 |
| Trapezium diagonals | AO/OC = BO/OD (parallel through O) | p. 7 |
6. Common mistakes
- Joining the wrong mark. It is the last mark on the ray that is joined to the far end B; the parallels go through the other marks.
- Unequal compass steps. The steps on the ray can be any length, but they must all be the same – do not change the compass opening.
- Wrong pair of sides. In "a parallel cuts the other two sides in the same ratio", compare top piece to bottom piece on each side: AP/PB = AQ/QC, not AP/AQ = PB/QC (that is also true, but mixing them up causes slips).
- "Half" is only for midpoints. A parallel through a point at 1 : 2 gives a third, not a half. Only the midline is half the third side.
- Median vs altitude. A median goes to the midpoint of the opposite side; an altitude is perpendicular to it. They coincide only in isosceles/equilateral cases.
- 2 : 1 from the vertex. AG = 2 GD, so AG is two-thirds of the median, GD one-third – not the other way round.
- Ratio vs lengths. Three parallels 2 cm and 3 cm apart cut a slanted line in the ratio 2 : 3 – the pieces are not 2 cm and 3 cm.
7. Flashcards പദങ്ങൾ
Click a card to flip it.
8. Scored quiz (10 questions)
Answer every question, then press Submit. Each wrong answer tells you which page to revisit.
9. Your study checklist
Pages you have marked as completed are ticked. Aim to tick them all before the model test.
If you scored 8 or more in the quiz and the checklist is complete, go to the Model Test Paper. Attempt it in 1½ hours without looking at the solutions.
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