In $$ riangle A D C, \angle D=90^{\circ} ; \angle C=45^{\circ} $$ $$ A D=8 $$ then find $$ A C $$.

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In $$ 	riangle A D C, \angle D=90^{\circ} ; \angle C=45^{\circ} $$ $$ A D=8 $$ then find $$ A C $$.

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In triangle $$ \triangle A D C $$, we have the following known conditions:

1. $$ \angle D = 90^\circ $$ (right angle)
2. $$ \angle C = 45^\circ $$
3. $$ A D = 8 $$

Since $$ \triangle A D C $$ is a right triangle and one of the angles is $$ 45^\circ $$, the other angle $$ \angle A $$ must also be $$ 45^\circ $$ because the angles in a triangle sum up to $$ 180^\circ $$. Therefore, $$ \triangle A D C $$ is a 45-45-90 triangle.

In a 45-45-90 triangle, the lengths of the legs are equal, and the length of the hypotenuse $$ h $$ is given by the formula:

$$
h = a\sqrt{2}
$$

where $$ a $$ is the length of each leg. In this case, $$ A D $$ and $$ A C $$ are the legs of the triangle.

Given that $$ A D = 8 $$, we can find $$ A C $$ as follows:

Since $$ A D = A C $$ in a 45-45-90 triangle, we have:

$$
A C = A D = 8
$$

Thus, the length of $$ A C $$ is $$ 8 $$.

Therefore, the final answer is:

$$
A C = 8
$$
$$ A C = 8 $$

In \( \triangle A D C, \angle D=90^{\circ} ; \angle C=45^{\circ} \) \( A D=8 \) then find \( A C \).

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