Week 11 · The isosceles triangle: base angles, the equilateral triangle, axis of symmetry and the 30°–60°–90° triangle · 3 lessons
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The 30°–60°–90° triangle and problems with isosceles triangles
Mathematics Prep 2 Term 1, Unit 3, Lesson 5
GoalUse the 30°–60°–90° triangle: the side opposite 30° is half the hypotenuse, and solve problems with heights and areas of isosceles and equilateral triangles.
New words
30°–60°–90° triangle · a right-angled triangle with acute angles 30° and 60°short leg · the leg opposite 30°; it is half the hypotenuselong leg · the leg opposite 60°; it is the short leg times √3height · the perpendicular distance from a vertex to the opposite side
Explanation
Cut an equilateral triangle in half along its axis of symmetry. You get a right-angled triangle with angles 30°, 60° and 90°. The short side is half of a side of the equilateral triangle, so the side opposite 30° is half the hypotenuse. If the hypotenuse is 10, the short leg is 5. The long leg comes from Pythagoras: √(10² − 5²) = √75 = 5√3, about 8.66. So the three sides are in the ratio 1 : √3 : 2. In the picture the sides are 5, 5√3 and 10. Remember which leg is which: the half rule is for the leg opposite 30°, the shortest side, not the longer leg. The rule works only in a right-angled triangle with a 30° angle. A right-angled isosceles triangle is different: its acute angles are 45° and 45°.
Worked examples
A 30°–60°–90° triangle has hypotenuse 12 cm. The short leg is 6 cm and the long leg is √(144 − 36) = √108 = 6√3, which is about 10.4 cm.
A ladder 8 m long makes an angle of 60° with the ground. Its top reaches a wall, so the triangle is 30°–60°–90° with the ladder as hypotenuse. The foot is opposite the 30° angle, so it is 8 ÷ 2 = 4 m from the wall. The height reached is √(64 − 16) = √48 = 4√3, about 6.9 m.
A roof face is an isosceles triangle with legs 10 m and vertex angle 120°. The height from the vertex bisects the angle into two 60° angles and meets the base at its midpoint, giving a 30°–60°–90° triangle with hypotenuse 10 m. The base angles are 30°, so the height, opposite 30°, is 5 m. Half the base is 5√3, so the base is 10√3, about 17.3 m. The area is ½ × 10√3 × 5 = 25√3, about 43.3 m².
Class activity
Half a triangle. Cut an equilateral triangle with sides of 10 cm and cut it along an axis. Measure the short side, the long side and the hypotenuse of one half. Check that the short side is half of the hypotenuse and the long side is about 8.7 cm.
Practice
1
A 30°–60°–90° triangle has hypotenuse 20 cm. Find the short leg.
10 cm.
2
The leg opposite 30° is 7 cm. Find the hypotenuse and the long leg.
Hypotenuse 14 cm; long leg √(196 − 49) = √147 = 7√3, about 12.1 cm.
3
An equilateral triangle has side 12 cm. Find its height and area. Give the height as a root.
Half-base 6, height √(144 − 36) = 6√3 cm. Area ½ × 12 × 6√3 = 36√3 cm² (about 62.4).
4
A right-angled isosceles triangle has legs 6 cm. Find its acute angles and hypotenuse.
45° and 45°; hypotenuse √72 = 6√2 cm, about 8.5 cm.
5
Dina says the long leg of a 30°–60°–90° triangle is half the hypotenuse. What did she mix up?
The half rule is for the short leg, opposite 30°. The long leg is about 0.87 of the hypotenuse.
