Noah and Brianna went to calculate the distance between their houses which are opposite sides of a water park. They mark a point, A, 120 m along the edge of the water park from Brianna's house. the measure NBA as 75° and BAN as 70°. Determine the distance between their houses.
To determine the distance between Noah and Brianna's houses, we can use trigonometry and the Law of Sines.
First, let's label the given information:
- The distance from Brianna's house to point A is 120 m.
- The measure of angle NBA is 75°.
- The measure of angle BAN is 70°.
Now, let's find the length of the side opposite angle NBA (which is Brianna's house to Noah's house) using the Law of Sines:
sin(NBA) / BA = sin(BAN) / AB
We know that sin(NBA) = sin(75°) and sin(BAN) = sin(70°). Let's substitute those values into the equation:
sin(75°) / BA = sin(70°) / AB
Next, we rearrange the equation to solve for BA (the distance between Brianna's house and the point A):
BA = (sin(75°) * AB) / sin(70°)
We still need to find the value of AB (the distance from Noah's house to point A). To find AB, we can use the fact that the three angles in a triangle sum to 180°:
NBA + BAN + ANB = 180°
75° + 70° + ANB = 180°
ANB = 180° - 75° - 70°
ANB = 35°
Now, we can use trigonometry again to find AB (the distance from Noah's house to point A) by using the Law of Sines:
sin(BAN) / AB = sin(ANB) / BA
sin(70°) / AB = sin(35°) / BA
Rearranging the equation to solve for AB:
AB = (sin(70°) * BA) / sin(35°)
Since we know that BA = 120 m from the problem statement, we can substitute that value into the equation:
AB = (sin(70°) * 120) / sin(35°)
Now we can calculate AB:
AB ≈ 159.78 m
Therefore, the distance between Noah and Brianna's houses is approximately 159.78 meters.