Math 1 · Unit 11 · Lesson 1 · part 3 of 5
The right triangle under the segment
The trick is to draw a right triangle under the slanted segment: one leg going across and one going up,...
The right triangle under the segment
The trick is to draw a right triangle under the slanted segment: one leg going across and one going up, meeting at a square corner. The legs are countable. The segment is the hypotenuse, the longest side, across from the right angle. And the Pythagorean theorem from 8th grade says, for any right triangle,
From (1, 2) to (4, 6): the leg across is 4 − 1 = 3, and the leg up is 6 − 2 = 4. So c2 = 32 + 42 = 9 + 16 = 25, and c = √25 = 5. The distance is 5.
Pick two points and watch the triangle appear. Then do the three steps.
Two more, by drawing. From (−3, −1) to (2, 5): across 2 − (−3) = 5, up 5 − (−1) = 6, so d = √5² + 6² = √61 ≈ 7.81. From (−2, 4) to (3, −1): across 5, down 5, so d = √5² + 5² = √50 ≈ 7.07.
d = √(x2 − x1)² + (y2 − y1)²
It is nothing but the Pythagorean theorem with the legs read off the grid. If you ever forget it, draw the triangle again.