Inequalities

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The symbol '>' means 'is greater than'. For example, if a is greater than b, we write a > b.

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The symbol '<' means 'is less than'. For example, if a is less than b, we write a < b.

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If two sides of a triangle are unequal, the greater side has the greater angle opposite to it. In other words, if AB > AC in triangle ABC, then ∠ACB > ∠ABC.

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If two angles of a triangle are unequal, the greater angle has the greater side opposite to it. In other words, if ∠A > ∠B in triangle ABC, then BC > AC.

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The perpendicular distance from the point to the line is the shortest distance. This is stated in Theorem 5: Of all the lines that can be drawn to a given straight line from a given point outside it, …

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The sum of the lengths of any two sides of a triangle is always greater than the third side. For triangle ABC: (i) AB + AC > BC, (ii) AB + BC > AC, and (iii) BC + AC > AB.

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The difference between the lengths of any two sides of a triangle is always less than the third side. For triangle ABC: (i) AB - AC < BC, (ii) AB - BC < AC, and (iii) BC - AC < AB.

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Using the triangle inequality theorem: AB - BC < AC < AB + BC. Therefore, 8 - 5 < AC < 8 + 5, which gives us 3 < AC < 13. So AC must be between 3 cm and 13 cm (exclusive).