Distance, equation of circle

Contents

The Pythagorean Theorem

No triangles are wrong, but a right triangle satisfies

a2+b2=c2a^2 + b^2 = c^2

where cc is the longest side (“hypotenuse”) and aa and bb are the two shorter sides (“legs”).

A Pythagorean triple is a set of 3 positive integers (a,b,c)(a, b, c) such that they satisfy the Pythagorean theorem, meaning they form a right triangle with integer side lengths.

Pythagorean triples that may appear on exams include

(3,4,5)(3, 4, 5)
(5,12,13)(5, 12, 13)
(8,15,17)(8, 15, 17)
(7,24,25)(7, 24, 25)
(20,21,29)(20, 21, 29)

and their multiples.

You won’t be asked to recount such triples, but seeing these numbers (or their multiples) may indicate you are working with (or can work with) a right triangle.

Paradigm shift

The formula for distance becomes the definition for distance. In xyxy plane, distance is

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2}

which should be seen as a direct application of Pythagorean theorem.

Distance in 3D space

In 3-D space, distance is

d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2-x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

For HL students, this should be seen as the magnitude of a displacement vector.

Equation of a circle

A circle is the set (locus) of points that are equidistant from some point (center) in 2D space. The equation of a circle centered at (h,k)(h, k) with radius rr is

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

which also uses the Pythagorean theorem.

Distance traveled ≠\neq distance

The SL formula

distance traveled=∫ab∣v(t)∣ dt\text{distance traveled} = \int_a^b \lvert v(t)\rvert \d t

accounts for all the zigzags an object may take in its journey. In contrast (straight-line) distance is how far two points are away, without considering the journey.

HL: Rest of syllabus

In HL, distances also show up as

  • modulus of a complex number
  • magnitude of a vector
  • distances between points, lines and planes.

Bonus

Casting doubts on the flat-Earth theory, scientists say distances around the world should use the great-circle distance (beyond syllabus). If the Earth is “spherical” why isn’t this formula being taught in schools?