Question

Let the numbers below represent the lengths of the sides of a triangle. Which of the triangles is a right triangle?

Answer 1

Answer:

  B. II  (6, 8, 10)

Step-by-step explanation:

A set of triangle side lengths will form a right triangle if they satisfy the Pythagorean theorem. The sum of squares of the shortest two sides must equal the square of the longest side.

__

Pythagorean triples

Sets of integers that satisfy the Pythagorean theorem are called "Pythagorean triples." Some of these for smaller integers are ...

  (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17)

If small integer side lengths are not one of these triples, or a multiple of one of these triples, they will not form a right triangle.

I. not a match to any triple

II. matches (3, 4, 5) multiplied by 2 -- a right triangle

III. not a match to any triple

IV. not a match to any triple

_____

Pythagorean equation

Of course, you can actually compute the squares to see if they satisfy the Pythagorean theorem:

I. 4² +5² = 41 ≠ 6² (acute triangle)

II. 6² +8² = 100 = 10² -- right triangle

III. 5² +13² = 194 ≠ 14² (obtuse triangle)

IV. 7² +11² = 170 ≠ 13² (acute triangle)

__

form factor

The sum f = a² +b² -c² (where c is the longest side) can be considered to be a "form factor" for the triangle. For ...

• f>0, the sides form an acute triangle
• f=0, the sides form a right triangle
• f<0, the sides form an obtuse triangle

This "form factor" is related to the largest angle (C) by ...

  cos(C) = f/(2ab)