Problem Solving

Question

rodney says if a triangle can be acute and isosceles at the same time, and another triangle can be equilateral ans isosceles at the same time, then every triangle can be described by two of the triangle words. So there must be some triangle that is scalene and isosceles and another that is right and obtuse. What are the limits to Rodney's conjecture? What combinations are possible and why? Which are not andwhy?

Answer 1

It is true that any triangle can be described using two triangle words, but one of the two words must be one that describes the sides and the other must be one that describes the angles. Therefore, a triangle can be isosceles and obtuse, scalene and right, equilateral and acute, etc., but not isosceles and scalene or right and acute.

Answer 2

Rodney says if a triangle can be acute and isosceles at the same time, and another triangle can be equilateral ans isosceles at the same time, then every triangle can be described by two of the triangle words. So there must be some triangle that is scalene and isosceles and another that is right and obtuse. What are the limits to Rodney's conjecture? What combinations are possible and why? Which are not and why?

Hi Karen. To answer this one, it helps if you know your definitions:

1. Triangle: 3-sided closed figure where the sum of the angles = 180°
2. Acute triangle: triangle with 3 angles less than 90°
3. Obtuse triangle: triangle with 1 angle greater than 90°
4. Right triangle: triangle with 1 angle equal to 90°
5. Isosceles triangle: triangle with (at least) 2 equal sides
6. Equilateral triangle: triangle with 3 equal sides
7. Scalene triangle: triangle with no equal sides

The first 3 (acute, obtuse and right) describe the angles of the triangle. The second 2 (isosceles, equilateral, scalene) the sides of the triangle.

The limits to Rodney's conjecture? (1) You have to follow the 180° rule, and (2) you can't have conflicting descriptors either. One other rule that you have to follow is that (3) if two angles in a triangle have the same measurements (example: both 45°), then their opposite sides also have the same measurements.

With that said, possible combinations include:

1. isosceles-equilateral (by definition)
2. acute equilateral (60°-60°-60° adds up to 180°, angles are less than 90°, rule (3) above)
3. acute isosceles (75°-75°-30° adds up to 180°, angles are less than 90°, rule (3) above)
4. acute scalene (85°-45°-50° adds up to 180°, angles are less than 90°, inverse of rule (3) above [all angles are different so all sides must be different])
5. obtuse isosceles (100°-40°-40° adds up to 180°, 1 angle is greater than 90°, rule (3)above)
6. obtuse scalene (100°-60°-20° adds up to 180°, 1 angle is greater than 90°, by definition, inverse of rule (3) above)
7. right isosceles (90°-45°-45° adds up to 180°, has a right angle, rule (3) above, pythagorian 1-1-square root of 2 triangle)
8. right scalene ((90°-36.87°-53.13° adds up to 180°, has a right angle, rule (3) above, pythagorian 3-4-5 triangle)

Impossible combinations:

1. obtuse equilateral (breaks rule (3) above)
2. right acute (by definition)
3. right obtuse (by definition)
4. right equilateral (breaks rule (3) above)

My question to you will be: Did I miss any?