Find the Side Length of A Right Triangle

Since we know 2 sides of this triangle, we will use the Pythagorean theorem to solve for x.

Substitute the two known sides into the Pythagorean theorem's formula:

$$ a^2 + b^2 = c^2 \\ 8^2 + 6^2 = x^2 \\ 100 = x^2 \\ x = \sqrt{100} \\ x = \boxed{10} $$

Since we know 1 side and 1 angle of this triangle, we will use sohcahtoa.

Set up an equation using a sohcahtoa ratio. Since we know the hypotenuse and want to find the side opposite of the 53° angle, we are dealing with sine

$$ sin(53) = \frac{ opposite}{hypotenuse} \\ sin(53) = \frac{ \red x }{ 12 } $$

Now, just solve the Equation:

$$ sin(53) = \frac{ \red x }{ 12 } \\ \red x = 12 \cdot sin (53) \\ \red x = \boxed{ 11.98} $$

Since we know 2 sides of this triangle, we will use the Pythagorean theorem to solve for side t.

Substitute the two known sides into the Pythagorean theorem's formula:

$$ a^2 + b^2 = c^2 \\ \red t^2 + 12^2 = 13^2 \\ \red t^2 + 144 = 169 \\ \red t^2 = 169 - 144 \\ \red t^2 = 25 \\ \red t = \boxed{5} $$

Since we know 1 side and 1 angle of this triangle, we will use sohcahtoa.

Set up an equation using the sine, cosine or tangent ratio Since we want to know the length of the hypotenuse, and we already know the side opposite of the 53° angle, we are dealing with sine.

$$ sin(67) = \frac{opp}{hyp} \\ sin(67) = \frac{24}{\red x} $$

Now, just solve the Equation:

$$ x = \frac{ 24}{ sin(67) } \\ x = 26.07 $$