GoalUse the equilateral triangle and the axis of symmetry: the axis of a segment, and the bisector, median and height that coincide in an isosceles triangle.
New words
equilateral triangle · all three sides are equal, and all three angles are 60°axis of symmetry · a line that folds a shape onto itselfaxis of a segment · the line perpendicular to the segment through its midpointequidistant · equally far from two points
Explanation
An equilateral triangle has all three sides equal if and only if all three angles are equal. Since the angles add up to 180°, each is 60°. Every equilateral triangle is isosceles (with any side as the base), but an isosceles triangle is equilateral only when its sides are all equal; one 60° angle in an isosceles triangle is enough, because then the other two angles are 60° as well. The axis of symmetry of a segment is the line perpendicular to it through its midpoint. A point lies on this axis if and only if it is the same distance from both ends. For an isosceles triangle ABC with AB = AC, the vertex A is on the axis of BC, so the line from A to the midpoint D of BC is one line that is at once the bisector of the vertex angle, the median to the base, and the height to the base. It is the triangle’s axis of symmetry. A scalene triangle has no axis, an isosceles triangle has one and an equilateral triangle has three, shown in the picture.
Worked examples
An isosceles triangle has AB = AC and the angle A = 60°. The base angles are (180° − 60°) ÷ 2 = 60° too, so all angles are 60° and the triangle is equilateral. If AB = 7 cm, the perimeter is 3 × 7 = 21 cm.
Isosceles Δ ABC has AB = AC = 13 cm and BC = 10 cm. The line AD from A to the midpoint D of BC is the median, the bisector and the height, so AD ⊥ BC and BD = 5 cm. Then AD = √(13² − 5²) = √144 = 12 cm. The area is ½ × 10 × 12 = 60 cm².
A point P is 6 cm from A and 6 cm from B, with AB = 10 cm. P lies on the axis of AB, so P is on the perpendicular through the midpoint M, with AM = 5 cm. The distance PM = √(36 − 25) = √11 cm, which is about 3.3 cm. A point Q that is 4 cm from A and 7 cm from B is not equally far, so it is not on the axis.
Class activity
Fold the axes. Cut an equilateral triangle from paper and fold it to find all its axes of symmetry. How many are there? Then cut an isosceles triangle that is not equilateral and count again. Check that the fold line meets the base at its midpoint and at a right angle.
Practice
1
An equilateral triangle has perimeter 36 cm. Find its side and each angle.
Side 12 cm; each angle 60°.
2
Isosceles Δ ABC has AB = AC = 17 cm and BC = 16 cm. Find the height from A.
BD = 8; height = √(289 − 64) = √225 = 15 cm.
3
Now find the area of that triangle.
½ × 16 × 15 = 120 cm².
4
A point is 5 cm from A and 5 cm from B. Is it on the axis of AB?
Yes: it is equally far from both ends.
5
How many axes of symmetry does a scalene triangle, an isosceles (not equilateral) triangle and an equilateral triangle have?
