Question

The medians of a triangle are the line segments from each vertex to the midpoint of the opposite side. Find the lengths of the medians of the triangle with vertices at A = (0,2), B = (6,2), and C = (4,6). The length of the median CD is ___?

Answer 1

The length of median CD of the triangle with vertices at A(0,2), B(6,2), and C(4,6) is found using the midpoint of AB and the Pythagorean theorem, resulting in approximately 4.1231 units.

Step-by-step explanation:

To find the length of median CD in the triangle with vertices A(0,2), B(6,2), and C(4,6), we first need to determine the midpoint of AB, which will be point D. Since AB is a horizontal line segment, the y-coordinate of the midpoint D will be the same as A and B which is 2. The x-coordinate of D is the average of the x-coordinates of A and B, which is (0+6)/2 = 3. Thus, D has coordinates (3,2).

Now, we apply the Pythagorean theorem, since triangle CDD' forms a right triangle, where DD' is the horizontal leg, CD is the median, and D'C is the vertical leg. The length of DD' is the difference between the x-coordinates of C and D which is |4-3| = 1. The length of D'C is the difference between the y-coordinates of C and D which is |6-2| = 4. Using the Pythagorean theorem, a2 + b2 = c2, where a and b are the legs of the right triangle, and c is the hypotenuse (median CD in this case).

Therefore, CD2 = 12 + 42 = 1 + 16 = 17, and the length of the median CD is the square root of 17, which is approximately 4.1231.