Acute, right or obtuse? Classifying a triangle by its sides
Mathematics Prep 2 Term 1, Unit 3, Lesson 2
GoalClassify a triangle by comparing c² with a² + b², also with roots and letters, and solve problems with ropes and kites.
New words
acute triangle · all three angles are less than 90°: c² < a² + b²right triangle · one angle is 90°: c² = a² + b²obtuse triangle · one angle is more than 90°: c² > a² + b²triangle inequality · the sum of two sides must be greater than the third
Explanation
Compare c², the square of the longest side, with a² + b². If c² = a² + b² the triangle is right-angled. If c² < a² + b² the longest side is ‘too short’ to open the angle fully, so all angles are acute. If c² > a² + b² the longest side is ‘too long’, so the angle opposite it is obtuse. For 5, 6, 7: 49 < 25 + 36 = 61, acute. For 4, 5, 7: 49 > 16 + 25 = 41, obtuse. The three lengths must first make a triangle: 3, 4, 8 cannot, because 3 + 4 < 8. The test works with roots and letters too: √3, √6 and 3 give 9 = 3 + 6, a right angle; and 3x, 4x, 5x give 25x² = 9x² + 16x², a right angle for every positive x.
Compare
Triangle
c² = a² + b²
right-angled
c² < a² + b²
acute
c² > a² + b²
obtuse
Worked examples
Classify the triangle with sides 4, 5, 7. First 4 + 5 = 9 > 7, so it is a triangle. c² = 49 and a² + b² = 16 + 25 = 41. Since 49 > 41, it is obtuse. Now 5, 6, 7: 49 < 61, so it is acute.
Is a triangle with sides √5, 2 and 3 right-angled? The longest side is 3 (since √5 ≈ 2.24). 3² = 9 and (√5)² + 2² = 5 + 4 = 9. They are equal, so yes. Is 3x, 4x, 5x right-angled? 9x² + 16x² = 25x² = (5x)². Yes.
A mast is 8 m high and a rope of 10 m runs from its top to the ground. The foot of the rope is √(10² − 8²) = √36 = 6 m from the mast. A kite on a 13 m string, 5 m to the side of the person, is √(169 − 25) = √144 = 12 m high. A school pitch 60 m by 80 m has a diagonal √(3600 + 6400) = 100 m.
Class activity
Sort the triangles. Make a table of five triples of lengths, for example 5-6-7, 6-8-10, 4-5-7, 7-9-11 and 8-15-17. Work out c² and a² + b² for each and say whether it is acute, right or obtuse. Check one by drawing it accurately.
Practice
1
Classify the triangle with sides 6, 8 and 9.
81 < 36 + 64 = 100, so it is acute.
2
Classify the triangle with sides 5, 12 and 14.
196 > 25 + 144 = 169, so it is obtuse.
3
Classify the triangle with sides 7, 24 and 25.
625 = 49 + 576, so it is right-angled.
4
Can 3 cm, 4 cm and 8 cm be the sides of a triangle?
No: 3 + 4 = 7 is less than 8.
5
Show that the triangle with sides 5x, 12x and 13x is right-angled (x > 0).
