A right triangle has legs a = 6 and b = 5. Find the hypotenuse c.

Step-by-step solution

1. 1

   Set up the Pythagorean theorem. The two given sides meet at the right angle, so they are the legs and $c$ is the hypotenuse:

   $a^2+b^2=c^2$

2. 2

   Substitute and square each leg.

   $6^2+5^2=c^2\quad\Longrightarrow\quad 36+25=c^2$

3. 3

   Add.

   $c^2=61$

4. 4

   Take the square root and try to simplify it. $\sqrt{61}$ can only be reduced if $61$ has a perfect-square factor. It does not — $61$ is prime — so the radical is already in simplest form:

   $c=\sqrt{61}$

5. 5

   Give a decimal if one is wanted. Since $7^2=49$ and $8^2=64$, the answer sits between $7$ and $8$:

   $c=\sqrt{61}\approx 7.81$

   Leave the answer as $\sqrt{61}$ whenever an exact value is requested; $7.81$ is only a rounding.

6. 6

   Check the triangle inequality. $7.81<6+5=11$ and $7.81>6-5=1$, so a triangle with these sides genuinely exists, and the hypotenuse is longer than either leg as it must be.