Quick idea: square roots and the Pythagorean Theorem

Square root: if n = a^2, then a is a square root of n. For example, 5 is a square root of 25 because 5^2 = 25. We write the positive square root as √25 = 5.

Pythagorean Theorem (for right triangles): For a right triangle with legs a and b and hypotenuse c,

a^2 + b^2 = c^2

That means: to find the missing side, square the known sides, add or subtract, then take a square root.

How to use it — step by step

1. If you want the hypotenuse c: c = √(a^2 + b^2). Square the two legs, add them, then take the square root.
2. If you want a missing leg a: a = √(c^2 - b^2). Square the hypotenuse and the known leg, subtract, then take the square root.
3. To check if a triangle with sides p, q, r is right: order them so r is the longest. If p^2 + q^2 = r^2, the triangle is right. If p^2 + q^2 < r^2 it's obtuse; if > it's acute.

Examples (step-by-step)

Example 1 — Find the hypotenuse

Legs 6 and 8. Hypotenuse c = √(6^2 + 8^2) = √(36 + 64) = √100 = 10.

Example 2 — Find a missing leg

Hypotenuse 13, one leg 5. Other leg a = √(13^2 - 5^2) = √(169 - 25) = √144 = 12.

Example 3 — Check whether 7, 24, 25 form a right triangle

7^2 + 24^2 = 49 + 576 = 625 and 25^2 = 625, so yes — it's a right triangle.

Pythagorean triples

A Pythagorean triple is three whole numbers (a, b, c) with a^2 + b^2 = c^2. Common ones:

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25

If (a, b, c) is a triple, then (ka, kb, kc) is also a triple (scale by k). For example, 6, 8, 10 is 2×(3,4,5).

Hint at generation: Euclid's formula (optional): for positive integers m > n, set a = m^2 - n^2, b = 2mn, c = m^2 + n^2. This often gives 'primitive' triples (not multiples of a smaller triple).

Distance in the coordinate plane (8.G.B.6)

To find the distance between points (x1, y1) and (x2, y2), draw a right triangle whose legs are |x2 - x1| and |y2 - y1|. Then apply the Pythagorean Theorem:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Example

Find distance between (1, 2) and (7, 10): legs are 6 and 8, so d = √(6^2 + 8^2) = √100 = 10.

Classifying triangles using the theorem (8.G.B.7)

If you have three side lengths, square the two shorter ones and add. Compare to the square of the longest side:

• If equal: right triangle.
• If less: obtuse (because a^2 + b^2 < c^2).
• If greater: acute (because a^2 + b^2 > c^2).

Pythagorean paths (8.G.B.8) — shortest routes on grids

Many problems give grid or map paths. Key idea: the straight-line distance across a rectangle is the diagonal, which is shorter than walking the two sides. Use the Pythagorean Theorem to compare.

Example: corner to corner

Start at one corner of a 6 by 8 rectangle, go to opposite corner. Walking along edges: 6 + 8 = 14. Going straight across (the diagonal): √(6^2 + 8^2) = 10. So diagonal is shorter.

Example: paths made of diagonal steps

If you can move diagonally (45° steps), a diagonal move across a 1-by-1 square has length √2. So to go from (0,0) to (5,5) using 5 diagonal moves you'd travel 5√2 ≈ 7.07, much shorter than the axis path of 10.

Worked practice problems

1. Find the hypotenuse of a right triangle with legs 9 and 12. (Answer: √(81+144)=√225=15.)
2. Hypotenuse is 17 and one leg is 8. Find the other leg. (Answer: √(289-64)=√225=15.)
3. Are 10, 24, 26 a right triangle? (Compute 10^2+24^2=100+576=676 and 26^2=676 → yes.)
4. Distance between (2, -1) and (7, 3). (Legs: 5 and 4 → distance √(25+16)=√41 ≈ 6.403.)
5. On a 3 by 4 grid, which is shorter: walk 3+4=7 along edges or go diagonal √(3^2+4^2)=5? (Answer: diagonal is shorter: 5.)

Tips to remember

• Always identify the hypotenuse (longest side) before using the theorem.
• Watch for common triples — they make arithmetic easy.
• Use the distance formula in coordinate problems — it is just the Pythagorean Theorem in disguise.
• When given paths, ask: can I form a right triangle? Then use a^2 + b^2 = c^2.

If you'd like, I can make a few step-by-step practice problems at Beast Academy difficulty, or explain any of these examples in more detail.