The figure shows the location of three points around a lake. The length of the lake, BC, is also shown.
(The figure is not drawn to scale.)
A picture of a right triangle ABC with right angle at B is shown. The length of the side AC is labeled as 6 miles. The length BC of the triangle is labeled as 2 miles. This length BC is also the length of an irregular gray shaded shape.
Which of the following choices is closest to the distance (in miles) between points A and B? (5 points)
3.46 miles
5.66 miles
6.32 miles
7.75 miles
We can solve this problem by using the Pythagorean theorem. In a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, we need to find the length of side AB.
Let's call the length of side AB "x". We know that the length of side AC is 6 miles, and the length of side BC is 2 miles. Applying the Pythagorean theorem, we have:
\(x^2 = 6^2 + 2^2\)
\(x^2 = 36 + 4\)
\(x^2 = 40\)
Taking the square root of both sides, we find:
\(x = \sqrt{40} \approx 6.32\)
So the distance between points A and B is closest to 6.32 miles. The closest answer choice is 6.32 miles.