Solve the Triangle tri{}{45}{9}{45}{}{90}

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 9 \\ a = \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

Find $b$ .

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The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.

$\cos (A) = \frac{adj}{hyp}$

Substitute the name of each side into the definition of the cosine function.

$\cos (A) = \frac{b}{c}$

Set up the equation to solve for the adjacent side, in this case $b$ .

$b = c \cdot \cos (A)$

Substitute the values of each variable into the formula for cosine.

$b = 9 \cdot \cos (45)$

Combine $9$ and $\frac{\sqrt{2}}{2}$ .

$b = \frac{9 \sqrt{2}}{2}$

Find the last side of the triangle using the Pythagorean theorem.

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Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

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

Solve the equation for $a$ .

$a = \sqrt{c^{2} - b^{2}}$

Substitute the actual values into the equation.

$a = \sqrt{{(9)}^{2} - {(\frac{9 \sqrt{2}}{2})}^{2}}$

Raise $9$ to the power of $2$ .

$a = \sqrt{81 - {(\frac{9 \sqrt{2}}{2})}^{2}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $\frac{9 \sqrt{2}}{2}$ .

$a = \sqrt{81 - \frac{{(9 \sqrt{2})}^{2}}{2^{2}}}$

Apply the product rule to $9 \sqrt{2}$ .

$a = \sqrt{81 - \frac{9^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Simplify the numerator.

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Raise $9$ to the power of $2$ .

$a = \sqrt{81 - \frac{81 {\sqrt{2}}^{2}}{2^{2}}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$a = \sqrt{81 - \frac{81 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$a = \sqrt{81 - \frac{81 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Combine $\frac{1}{2}$ and $2$ .

$a = \sqrt{81 - \frac{81 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$a = \sqrt{81 - \frac{81 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Divide $1$ by $1$ .

$a = \sqrt{81 - \frac{81 \cdot 2}{2^{2}}}$

Evaluate the exponent.

$a = \sqrt{81 - \frac{81 \cdot 2}{2^{2}}}$

Reduce the expression by cancelling the common factors.

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Raise $2$ to the power of $2$ .

$a = \sqrt{81 - \frac{81 \cdot 2}{4}}$

Multiply $81$ by $2$ .

$a = \sqrt{81 - \frac{162}{4}}$

Cancel the common factor of $162$ and $4$ .

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Factor $2$ out of $162$ .

$a = \sqrt{81 - \frac{2 (81)}{4}}$

Cancel the common factors.

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Factor $2$ out of $4$ .

$a = \sqrt{81 - \frac{2 \cdot 81}{2 \cdot 2}}$

Cancel the common factor.

$a = \sqrt{81 - \frac{2 \cdot 81}{2 \cdot 2}}$

Rewrite the expression.

$a = \sqrt{81 - \frac{81}{2}}$

To write $81$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = \sqrt{81 \cdot \frac{2}{2} - \frac{81}{2}}$

Combine $81$ and $\frac{2}{2}$ .

$a = \sqrt{\frac{81 \cdot 2}{2} - \frac{81}{2}}$

Combine the numerators over the common denominator.

$a = \sqrt{\frac{81 \cdot 2 - 81}{2}}$

Simplify the numerator.

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Multiply $81$ by $2$ .

$a = \sqrt{\frac{162 - 81}{2}}$

Subtract $81$ from $162$ .

$a = \sqrt{\frac{81}{2}}$

Rewrite $\sqrt{\frac{81}{2}}$ as $\frac{\sqrt{81}}{\sqrt{2}}$ .

$a = \frac{\sqrt{81}}{\sqrt{2}}$

Simplify the numerator.

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Rewrite $81$ as $9^{2}$ .

$a = \frac{\sqrt{9^{2}}}{\sqrt{2}}$

Pull terms out from under the radical.

$a = \frac{| 9 |}{\sqrt{2}}$

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$a = \frac{9}{\sqrt{2}}$

Multiply $\frac{9}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a = \frac{9}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

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Multiply $\frac{9}{\sqrt{2}}$ and $\frac{\sqrt{2}}{\sqrt{2}}$ .

$a = \frac{9 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$a = \frac{9 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$a = \frac{9 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a = \frac{9 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$a = \frac{9 \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$a = \frac{9 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$a = \frac{9 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$a = \frac{9 \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$a = \frac{9 \sqrt{2}}{2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$a = \frac{9 \sqrt{2}}{2}$

Evaluate the exponent.

$a = \frac{9 \sqrt{2}}{2}$

These are the results for all angles and sides for the given triangle.

$A = 45$

$B = 45$

$C = 90$

$a = \frac{9 \sqrt{2}}{2}$

$b = \frac{9 \sqrt{2}}{2}$

$c = 9$

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