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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 midpoint theorem
The line joining the midpoints 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 midpoint theorem
The line drawn from the midpoint 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:
 All pairs of corresponding angles are equal, and
 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 rightangled 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.
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