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 .
Anyone please help. I really don't know how to solve this questione
draw the altitude from A to form a right-angled triangle.
Cos C = 4/16 = 1/4
In triangle BCD, by the cosine law ...
BD^2 =8^2 + 8^2 - 2(8)(8)cosC
= 64 +64 - 128(1/4)
= 160
BD = √160 = 4√10
Where does the centroid G enter the picture?
To find BD, we can use the properties of isosceles triangles and the centroid.
1. In an isosceles triangle, the medians from the base angles are equal in length.
2. The centroid divides each median into two segments, with the distance from the centroid to the vertex being twice the distance from the centroid to the midpoint of the opposite side.
Since D is the midpoint of AC, AD = DC = 8.
Let's calculate the length of BD:
1. Apply the Pythagorean theorem to find the height of the isosceles triangle ABC.
Let h be the height of the triangle.
Using the length of the base BC = 8 and the lengths of the equal sides AB = AC = 16:
h^2 = 16^2 - (8/2)^2
h^2 = 256 - 16
h^2 = 240
h = √240
h ≈ 15.49
2. The length of AD can be calculated using the Pythagorean theorem in triangle ADB.
Using the height h ≈ 15.49 and AD = 8:
BD^2 + h^2 = AB^2
BD^2 = AB^2 - h^2
BD^2 = 16^2 - 15.49^2
BD^2 = 256 - 240
BD^2 ≈ 16
BD ≈ √16
BD ≈ 4
Therefore, the length of BD is approximately 4 units.
To find BD, we can first find the length of AD and DC, and then use the property that D is the midpoint of side AC.
Since ABC is an isosceles triangle, it means that AB and AC are equal. Therefore, we have AB = AC = 16.
We are given that BC = 8, and since ABC is isosceles, BC is also equal to AB and AC.
To find AD, we need to find the length of AC. Since D is the midpoint of AC, we can split AC into two equal halves: AD and DC.
AC = AD + DC
Since AC = AB = BC = 16, we have:
16 = AD + DC
Since D is the midpoint of AC, AD = DC. Let's say AD = x, then DC = x.
16 = x + x
16 = 2x
Divide both sides of the equation by 2:
8 = x
So, AD = DC = x = 8.
Now to find BD, we can use the fact that G is the centroid of triangle ABC. The centroid divides each median in the ratio 2:1.
Since D is the midpoint of AC, AD and DC are the medians of triangle ABC.
Therefore, BD is divided into two segments: GD and DG, such that GD = 2DG.
We already found AD = 8, so DG = AD/3.
DG = 8/3
Therefore, GD = 2 * (8/3) = 16/3.
So, BD = GD + DG = (16/3) + (8/3) = 24/3 = 8.
Therefore, the length of BD is 8.