Lesson 7 of 10 · 7 min
Testing collinearity and right angles
NCERT §11.4, Examples 4-5
The club plans a straight flight through three waypoints A(1, 0, 1), B(2, 2, 3) and C(4, 6, 7). Are they really on one line?
The lesson in notes
In short
Three points lie on one line (are collinear) exactly when the largest of the three distances between them equals the sum of the other two.
P(−2, 3, 5), Q(1, 2, 3), R(7, 0, −1): PQ² = 9 + 1 + 4 = 14, QR² = 36 + 4 + 16 = 56 and PR² = 81 + 9 + 36 = 126. So PQ = √14, QR = 2√14 and PR = 3√14, and PQ + QR = PR: the points are collinear.
Simplify the surds before adding. √14, √56 and √126 hide the pattern; √14, 2√14 and 3√14 show it at once.
For a right angle, compare squares: the triangle is right angled exactly when the square of the longest side equals the sum of the squares of the other two (Pythagoras and its converse).
A(3, 6, 9), B(10, 20, 30), C(25, −41, 5): AB² = 49 + 196 + 441 = 686, BC² = 225 + 3721 + 625 = 4571 and CA² = 484 + 2209 + 16 = 2709. The longest is BC², and 686 + 2709 = 3395 ≠ 4571, so the triangle is not right angled.
Work with squared distances for right-angle and isosceles tests: they stay whole numbers, and no square root is needed.