Lesson Guide
Explanation
The distance formula is Pythagoras written with coordinates. From A at negative two, negative one to B at four, seven, the run is x two minus x one: four minus negative two, which is six. The rise is y two minus y one: seven minus negative one, which is eight. Square both and add: thirty six plus sixty four is one hundred. The distance is the square root of one hundred, which is ten. So the distance is the square root of the run squared plus the rise squared.
Your Turn
Find the distance from (1, −3) to (4, 1). Check: 32 + 42 = 25, so d = 5.
Watch Out
Subtract, then square. (−3)2 is +9, never −9.
Explanation
The answer is not always a whole number. From negative three, four to one, negative two, the differences are four and negative six. Squaring gives sixteen and thirty six, which add to fifty two. Fifty two is not a perfect square, so the exact distance is root fifty two. You can tell it is between seven and eight, because fifty two is between forty nine and sixty four. If you swap the two points, the differences change sign, but their squares do not, so the order does not matter.
Your Turn
Find the distance from (0, 5) to (2, 1). Check: 22 + (−4)2 = 20, so d = √20.
Watch Out
√(16 + 36) is not √16 + √36 = 10. Add first, then take the square root.
Explanation
To test whether a triangle is isosceles, find its three side lengths and look for two that are equal. You can compare the squares and skip the square roots. For P at one, one, Q at four, seven and R at seven, one: P Q squared is nine plus thirty six, which is forty five. Q R squared is also nine plus thirty six, forty five. P R squared is thirty six. P Q equals Q R, so the triangle is isosceles.
Your Turn
Is A(0, 0), B(3, 4), C(7, 1) isosceles? Check: AB = 5 and BC = 5, so yes.
Watch Out
Equal squared lengths mean equal sides, so you do not need the square roots to compare them.
Explanation
To test for a right angle, use Pythagoras backwards. Find the three squared lengths. For A at negative one, one, B at three, three and C at zero, nine: A B squared is twenty, B C squared is forty five, and A C squared is sixty five. The longest side is A C. Twenty plus forty five is sixty five, so the triangle is right-angled, and the right angle is at B, the corner opposite the longest side.
Your Turn
Is (0, 0), (4, 0), (4, 3) right-angled? Check: 16 + 9 = 25, so yes, at (4, 0).
Watch Out
The two SMALLER squares must add to the LARGEST. The right angle is opposite the longest side.
Explanation
Sometimes the distance is given and a coordinate is missing. A is one, two, B is k, five, and A B is five. Square the distance and use the formula: k minus one, all squared, plus three squared, equals twenty five. Subtract nine: k minus one, all squared, equals sixteen. A number squared is sixteen when it is four or negative four, so k is five or negative three. Both work: B can sit to the right of A or to the left. Quiz and exercises now.
Your Turn
P(0, k) is 13 units from Q(12, 0). Check: k2 + 144 = 169, so k = 5 or k = −5.
Watch Out
Do not drop the negative square root. (k − 1)2 = 16 has two solutions, so there are two points.
📌 Key Points — this lesson in a nutshell
- Distance formula: d = √((x2 − x1)2 + (y2 − y1)2). It is Pythagoras, with the run and the rise as the short sides.
- The order of the points does not matter: squaring removes any minus sign.
- Compare squared lengths to test a triangle. Two equal: isosceles. The two smaller adding to the largest: right-angled.
- For a missing coordinate, write d2 as an equation. There are usually two answers.
Step 1: Introduction
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