Question

In isosceles triangle ABC, BD is the bisector of vertex angle B. If AB = 2x + 2y, AD = x + 2y, DC = 4x, and BC = 10, find the values of AB, AD, and AC.

Answer 1

To find the values of AB, AD, and AC in the isosceles triangle ABC, we can use the given information and properties of isosceles triangles. By applying the angle bisector theorem, we can create an equation and solve for x and y. Substituting the values of x and y in the given expressions, we can find the values of AB, AD, and AC.

Step-by-step explanation:

To find the values of AB, AD, and AC in the isosceles triangle ABC, we can use the given information and properties of isosceles triangles. Firstly, since BD is the bisector of vertex angle B, we can use the angle bisector theorem to find the values of AD and DC. According to the theorem, the ratio of the lengths of the segments created by the angle bisector is equal to the ratio of the lengths of the opposite sides. In this case, we have AD/DC = AB/BC. By substituting the given values, we get (x+2y)/4x = (2x+2y)/10. Simplifying this equation, we can solve for x and y, which will give us the values of AB, AD, and AC.

By cross-multiplying and simplifying, we get 5(x+2y) = 2(2x+2y). Expanding both sides, we get 5x + 10y = 4x + 4y. Simplifying further, we get x = -6y.

So, the values of x and y are x = -6y and y = -x/6. Substituting the values of x and y in the given expressions for AB, AD, and AC, we can find the values of AB, AD, and AC.
AB = 2(-6y) + 2y = -12y + 2y = -10y.
AD = (-6y) + 2y = -4y.
AC = AB + BC = -10y + 10 = 10 - 10y.