In the figure, if AC¯¯¯¯¯¯¯¯≅CE¯¯¯¯¯¯¯¯ and B is the midpoint of AC¯¯¯¯¯¯¯¯. CD = 2 and AB = 3.
To find the length of BD¯¯¯¯¯¯¯¯, we can use the properties of similar triangles.
First, since AC¯¯¯¯¯¯¯¯≅CE¯¯¯¯¯¯¯¯ and B is the midpoint of AC¯¯¯¯¯¯¯¯, we know that AB¯¯¯¯¯¯¯¯=BC¯¯¯¯¯¯¯¯.
Given that AB¯¯¯¯¯¯¯¯=3, this means that BC¯¯¯¯¯¯¯¯=3 as well.
Now, since B is the midpoint of AC¯¯¯¯¯¯¯¯, we can say that AD¯¯¯¯¯¯¯¯=DC¯¯¯¯¯¯¯¯.
Given that CD¯¯¯¯¯¯¯¯=2, this means that AD¯¯¯¯¯¯¯¯=2 as well.
Now, we have enough information to find BD¯¯¯¯¯¯¯¯ using the Pythagorean theorem.
In a right triangle ABD, we have AB¯¯¯¯¯¯¯¯=3 and AD¯¯¯¯¯¯¯¯=2. We want to find the length of BD¯¯¯¯¯¯¯¯.
Using the Pythagorean theorem, we have:
(AB¯¯¯¯¯¯¯¯)² + (AD¯¯¯¯¯¯¯¯)² = (BD¯¯¯¯¯¯¯¯)²
Substituting the given values, we get:
(3²) + (2²) = (BD¯¯¯¯¯¯¯¯)²
Simplifying, we get:
9 + 4 = (BD¯¯¯¯¯¯¯¯)²
13 = (BD¯¯¯¯¯¯¯¯)²
To solve for BD¯¯¯¯¯¯¯¯, we take the square root of both sides:
√13 = BD¯¯¯¯¯¯¯¯
Therefore, BD¯¯¯¯¯¯¯¯ is equal to the square root of 13, or approximately 3.606.