To determine the distance between Noah and Brianna's houses, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant.
In this case, we can set up the following ratios:
sin(BAN) / BA = sin(NBA) / AN
We are given that angle BAN measures 70 degrees, so we can substitute the values:
sin(70) / BA = sin(75) / AN
Next, we can find the value of sin(70) and sin(75). Using a calculator, we obtain:
sin(70) ≈ 0.9397
sin(75) ≈ 0.9659
Substituting these values into the equation, we have:
0.9397 / BA = 0.9659 / AN
Now, to simplify the equation, we can cross-multiply:
0.9397 * AN = 0.9659 * BA
Divide both sides of the equation by 0.9397 to solve for AN:
AN = (0.9659 * BA) / 0.9397
We are given that point A is 120m along the edge of the water park from Brianna's house. Therefore, BA = 120m.
Substituting this value into the equation, we have:
AN = (0.9659 * 120) / 0.9397
Using a calculator to perform the calculation, we get:
AN ≈ 124.1888
Hence, the distance between Noah and Brianna's houses is approximately 124.1888 meters.