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8.7 Summary (EMCJJ)

  • A ratio describes the relationship between two quantities which have the same units.

    \[x:y \quad \text{ or } \quad \frac{x}{y} \quad \text{ or } \quad x \enspace \text{ to } \enspace y\]
  • If two or more ratios are equal to each other \(\left( \frac{m}{n} = \frac{p}{q} \right)\), then \(m\) and \(n\) are in the same proportion as \(p\) and \(q\).

  • A polygon is a plane, closed shape consisting of three or more line segments.

  • Triangles with equal heights have areas which are proportional to their bases.

  • Triangles with equal bases and between the same parallel lines are equal in area.

  • Triangles on the same side of the same base and equal in area lie between parallel lines.

  • A line drawn parallel to one side of a triangle divides the other two sides of the triangle in the same proportion.

  • Converse: proportion theorem

    If a line divides two sides of a triangle in the same proportion, then the line is parallel to the third side.

  • Special case: the mid-point theorem

    The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side.

    If \(AB = BD\) and \(AC = CE\), then \(BC \parallel DE\) and \(BC = \frac{1}{2}DE\).

  • Converse: the mid-point theorem

    The line drawn from the mid-point of one side of a triangle parallel to another side, bisects the third side of the triangle.

    If \(AB = BD\) and \(BC \parallel DE\), then \(AC = CE\).

  • Polygons are similar if they are the same shape but differ in size. One polygon is an enlargement of the other.

  • Two polygons with the same number of sides are similar when:

    1. All pairs of corresponding angles are equal, and
    2. All pairs of corresponding sides are in the same proportion.
  • If two triangles are equiangular, then the triangles similar.

  • Triangles with sides in proportion are equiangular and therefore similar.

  • The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

  • Converse: theorem of Pythagoras

    If the square of one side of a triangle is equal to the sum of the squares of the other two sides of the triangle, then the angle included by these two sides is a right angle.