S2 Geometry

Question 1

Pythagoras’ Theorem by definition

Answer 1

The sum of squares of two legs of a right triangle is equal to the square of its hypotenuse.
a² + b² = c² (Pyth. Thm)

Question 2

Converse of Pythagoras’ Theorem

Answer 2

If the sum of squares of two sides of a triangle is equal to the square of its longest side, the triangle is right with the angle opposite the longest side being right.
If a² + b² = c², then angle opposite c is an right angle. (converse of Pyth. Thm)

Question 3

Definition of isosceles triangles

Answer 3

A triangles where two sides are of equal length. (legs)

Question 4

Reasons to prove isosceles triangles

Answer 4

1. prove that two legs of the triangle are equal
2. prove that base angles of the triangle are equal, then by (sides opp. equal <s) prove that the legs are equal

Question 5

Properties of isosceles triangles

Answer 5

1. Angles opposite the equal sides are equal in size. (base<s, isos. ∆)
2. If a line is drawn from the vertex angle to the base,
   a) if the line bisects the base, then it bisects the vertex angle and is perpendicular to the base. (prop. of isos. ∆)
   b) if the line bisects the vertex angle, then it perpendicularly bisects the base. (prop. of isos. ∆)
   c) if the line is perpendicular to the base, then it bisects the base and the vertex angle (prop. of isos. ∆)

note that prop. of isos. ∆ can only be used when we already know the triangle is isosceles.