Pythagorean Theorem - Find Missing Side - Square Roots worksheet ... - Free Printable

To find the missing side of a right triangle, we use the Pythagorean theorem: $a^2 + b^2 = c^2$.
* $a$ and $b$ are the legs (the sides that make the L-shape or right angle).
* $c$ is the hypotenuse (the longest side, opposite the right angle).

If you are looking for the hypotenuse ($c$), you add the squares of the legs: $c = \sqrt{a^2 + b^2}$.
If you are looking for a leg ($a$ or $b$), you subtract the square of the known leg from the square of the hypotenuse: $a = \sqrt{c^2 - b^2}$.

Here is the step-by-step solution for each problem:

1. Find the hypotenuse
* Legs: 6 in and 4 in.
* Formula: $c = \sqrt{6^2 + 4^2}$
* Calculate squares: $36 + 16 = 52$
* Square root: $\sqrt{52}$
* Simplify: 52 can be split into $4 \times 13$. Since 4 is a perfect square, we take its root (2) out.
* Result: $2\sqrt{13}$

2. Find the hypotenuse
* Legs: 6 cm and 7 cm.
* Formula: $c = \sqrt{6^2 + 7^2}$
* Calculate squares: $36 + 49 = 85$
* Square root: $\sqrt{85}$
* Simplify: 85 ends in 5, so it is divisible by 5 ($5 \times 17$). Neither number is a perfect square, so it cannot be simplified further.
* Result: $1\sqrt{85}$ (or just $\sqrt{85}$)

3. Find the missing leg
* Hypotenuse: 13 m. Known leg: 3 m.
* Formula: $a = \sqrt{13^2 - 3^2}$
* Calculate squares: $169 - 9 = 160$
* Square root: $\sqrt{160}$
* Simplify: Look for perfect squares in 160. $16 \times 10 = 160$. The square root of 16 is 4.
* Result: $4\sqrt{10}$

4. Find the missing leg
* Hypotenuse: 20 in. Known leg: 10 in.
* Formula: $a = \sqrt{20^2 - 10^2}$
* Calculate squares: $400 - 100 = 300$
* Square root: $\sqrt{300}$
* Simplify: Look for perfect squares in 300. $100 \times 3 = 300$. The square root of 100 is 10.
* Result: $10\sqrt{3}$

5. Find the hypotenuse
* Legs: 8 cm and 12 cm.
* Formula: $c = \sqrt{8^2 + 12^2}$
* Calculate squares: $64 + 144 = 208$
* Square root: $\sqrt{208}$
* Simplify: Look for perfect squares in 208. $16 \times 13 = 208$. The square root of 16 is 4.
* Result: $4\sqrt{13}$

Final Answer:
1. Answer = 2 $\sqrt{13}$ in
2. Answer = 1 $\sqrt{85}$ cm
3. Answer = 4 $\sqrt{10}$ m
4. Answer = 10 $\sqrt{3}$ in
5. Answer = 4 $\sqrt{13}$