18.7 Chech Your Underntandions $$ riangle R B N $$ is a triangle where $$ m \angle R B N=90^{\circ}, m \angle R N B=60^{\circ} $$ and $$ R N=8 \sqrt{2} $$ What is the length of $$ B N P $$ 4 $$ 4 \sqrt{2} $$ $$ 4 \sqrt{6} $$ $$ 2 \sqrt{5} $$ 1 Brat

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18.7 Chech Your Underntandions $$ 	riangle R B N $$ is a triangle where $$ m \angle R B N=90^{\circ}, m \angle R N B=60^{\circ} $$ and $$ R N=8 \sqrt{2} $$ What is the length of $$ B N P $$ 4 $$ 4 \sqrt{2} $$ $$ 4 \sqrt{6} $$ $$ 2 \sqrt{5} $$ 1 Brat

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

**Given:**
- Triangle $$ \triangle RBN $$ with a right angle at $$ B $$ ($$ \angle RBN = 90^\circ $$).
- $$ \angle RNB = 60^\circ $$, which makes $$ \angle BRN = 30^\circ $$ (since the sum of angles in a triangle is $$ 180^\circ $$).
- The length of side $$ RN = 8\sqrt{2} $$.

**Objective:**
- Find the length of $$ BN $$.

**Solution:**

In a $$ 30^\circ-60^\circ-90^\circ $$ triangle, the sides are in the ratio $$ 1 : \sqrt{3} : 2 $$, where:
- The side opposite the $$ 30^\circ $$ angle ($$ BN $$) is the shortest side.
- The side opposite the $$ 60^\circ $$ angle ($$ RB $$) is $$ \sqrt{3} $$ times the shortest side.
- The hypotenuse ($$ RN $$) is twice the shortest side.

Given that $$ RN = 8\sqrt{2} $$, we can set up the following relationship based on the ratio:
$$
2 \times (\text{shortest side}) = 8\sqrt{2}
$$
$$
\text{shortest side} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}
$$

So, the length of $$ BN $$ is $$ 4\sqrt{2} $$.

**Answer:**
$$ 4 \sqrt{2} $$
The length of $$ BN $$ is $$ 4 \sqrt{2} $$.

18.7 Chech Your Underntandions
is a triangle where and What is the length of
4

1 Brat

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18.7 Chech Your Underntandions is a triangle where and What is the length of 4 1 Brat