If a triangle has sides of lengths 3, 4, and 5 units, then it is a right triangle. All right triangles have an area equal to one half the product of the two smaller side lengths.

Which statement is valid, based on deductive reasoning?

Question

If a triangle has sides of lengths 3, 4, and 5 units, then it is a right triangle. All right triangles have an area equal to one half the product of the two smaller side lengths. 

Which statement is valid, based on deductive reasoning?

anonymous · Answer

A) A polygon has an area of 6 square units if it is a triangle.
	 
B) A triangle has a perimeter of 12 units if its sides are consecutive integers.
	 
C) If a triangle has side lengths 3, 4, and 5 units, then its area is 6 square units.
	 
D) If a triangle has side lengths 4, 5, and 6 units, then its area is 10 square units.

whpalmer4 · Answer

A triangle with sides 3,4,5 is a right triangle because 3,4,5 is a Pythagorean triple satisfying the relationship $a^2+b^2=c^2$ and that is only true for a right triangle.

Furthermore, all triangles have an area equal to one-half the product of their base and height, and in a right triangle, the sides which are not the hypotenuse are equal to the base and height.

whpalmer4 · Answer

We can rule out all but the correct answer with a modicum of thought.  For example, A is obviously incorrect, as triangles can be made with any area we specify., and B fails via similar argument about the length of the sides.