Carly found the following coordinate geometry proof in an old textbook.
Proof:Let l1 and l2 be perpendicular lines. Let l1 have slope m and l2 have slope n, where neither m nor n is zero. Let (a, b) be the point of intersection of lines l1 and l2 where a and b are real numbers.
By the definition of slope as change in y divided by change in x, l1 must pass through the point (a + 1, b + m). Similarly, l2 must pass through the point (a + 1, b + n). Since l1 ⊥ l2, these two points along with point (a, b) form a right triangle with right angle at (a, b). By the Pythagorean theorem, the following is true.
|n − m|2 = (1 + m2) + (1 + n2)
n2 − 2mn + m2 = 2 + m2 + n2
n2 − 2mn = 2 + n2
−2mn = 2
mn = −1
m = −
What is the conclusion of the proof that Carly found?
Responses
If two lines are perpendicular, then their slopes are negative reciprocals.
If two lines have slopes that are negative reciprocals, then they are perpendicular.
If a triangle is a right triangle, then the lengths of its sides stand in the relationship a2 + b2 = c2.
If the lengths of the sides of a triangle stand in the relationship a2 + b2 = c2, then it is a right triangle.