ABC is an isosceles triangle. If AB =AC =16, BC=8. D is the midpoint of side AC , and G is the centroid of triangle ABC , find BD .

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To find BD, we can use the properties of isosceles triangles and the definition of midpoints. Here's how you can find the length of BD step-by-step:

1. In an isosceles triangle, the two congruent sides are the legs and the third side is the base. In triangle ABC, AB = AC = 16, and BC = 8.

2. The midpoint of a side of a triangle divides the side into two equal parts. Given that D is the midpoint of AC, we can conclude that AD = DC = 8 (half of 16).

3. The centroid G divides each median into two sections, with the ratio 2:1. Since D is the midpoint of AC, we know that AG = 2GD.

4. To find GD, we can set up an equation using the ratios. Let's say GD = x. Then AG = 2x.

5. Since the centroid divides the medians into two sections, we know that the length of BD is equal to DG. Therefore, BD = GD = x.

6. Now, let's calculate the lengths of AG and GD using their relationship. AG = 2x and GD = x. Since AG + GD = AD, we can set up an equation: 2x + x = 8.

7. Simplifying the equation gives us 3x = 8. Dividing both sides by 3 gives x = 8/3.

8. Therefore, GD = BD = x = 8/3. So, the length of BD is 8/3 units.

To find the length of BD, we can use the properties of an isosceles triangle, the midpoint formula, and the centroid formula.

First, let's label the points:
- A, B, C are the vertices of the triangle ABC.
- D is the midpoint of side AC.
- G is the centroid of the triangle ABC.

Given information:
- AB = AC = 16 (isosceles triangle)
- BC = 8
- D is the midpoint of AC

Step 1: Find the length of BD.
- Since BD is a median of triangle ABC, it divides side AC into two equal segments.
- Using the midpoint formula, we can find the coordinates of point D.
- Let's assume point A is (0,0), and point C is (16, 0).
- The x-coordinate of point D is the average of the x-coordinates of A and C, which is (0 + 16) / 2 = 8.
- The y-coordinate of point D is the average of the y-coordinates of A and C, which is (0 + 0) / 2 = 0.
- Thus, the coordinates of point D are (8, 0).

Step 2: Find the coordinates of point G (the centroid).
- The centroid of the triangle ABC is the point of intersection of the medians.
- The coordinates of the centroid (G) can be found using the following formula:
G = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3)
- Let the coordinates of A be (x1, y1) = (0, 0).
- Let the coordinates of B be (x2, y2).
- Let the coordinates of C be (x3, y3) = (16, 0).
- We know that the centroid lies on the median BD, so its x-coordinate is the same as that of point D, i.e., 8.
- Now we can substitute the coordinates into the formula and solve for y2:
(8, (0 + y2 + 0) / 3) = ((0 + x2 + 16) / 3, (0 + y2 + 0) / 3)
(8, y2/3) = (x2 + 16) / 3, y2/3
3 * 8 = x2 + 16
24 - 16 = x2
x2 = 8
- Therefore, the coordinates of point B are (8, y2).
- Since B is the midpoint of AC, its y-coordinate is halfway between the y-coordinates of A and C:
y2 = (0 + 0) / 2
y2 = 0
- Thus, the coordinates of point B are (8, 0).

Step 3: Find the length of BD.
- We can now use the distance formula to find the length of BD.
- The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Let (x1, y1) be the coordinates of point B (8, 0).
- Let (x2, y2) be the coordinates of point D (8, 0).
- Substituting the values into the formula:
d = sqrt((8 - 8)^2 + (0 - 0)^2)
= sqrt(0 + 0)
= sqrt(0)
= 0

Therefore, the length of BD is 0.