Isosceles Triangle ABC

To find the measure of angle BAC in an isosceles triangle ABC, we need to consider the properties of isosceles triangles. An isosceles triangle is a triangle with two sides of equal length. In this case, triangle ABC has two equal sides, with one side measuring 12 cm and the other side measuring 19 cm.

Properties of Isosceles Triangles
1. Base Angles: The two angles opposite the equal sides are called the base angles. In an isosceles triangle, the base angles are equal.
2. Vertex Angle: The angle between the two equal sides is called the vertex angle. In an isosceles triangle, the vertex angle is opposite the base.

Finding Angle BAC
Since the triangle is isosceles, we know that the base angles are equal. Let's denote the base angles as angle B and angle C. The vertex angle, angle A, is opposite the base. To find the measure of angle BAC, we need to determine the value of either angle B or angle C.

Since the triangle ABC is isosceles with one side measuring 12 cm and the other side measuring 19 cm, we can conclude that angles B and C are equal.

Using the Triangle Angle Sum Theorem
The sum of the three angles in any triangle is always 180 degrees. We can use this information to find the measure of angle BAC.

Let's denote angle BAC as x. Since angles B and C are equal, we can represent them as y. Applying the Triangle Angle Sum Theorem, we have:

x + y + y = 180

Simplifying the equation, we get:

x + 2y = 180

Substituting Known Values
We know that one side of the triangle measures 12 cm, while the other side measures 19 cm. Using this information, we can apply the Law of Cosines to find the value of y.

The Law of Cosines states that in any triangle with sides a, b, and c, and angle A opposite side a, we have:

a^2 = b^2 + c^2 - 2bc*cos(A)

In our case, side a is 12 cm, and sides b and c are both 19 cm. Substituting these values into the equation, we get:

12^2 = 19^2 + 19^2 - 2*19*19*cos(y)

Simplifying the equation further, we have:

144 = 2*19^2 - 2*19^2*cos(y)

144 = 2*19^2(1 - cos(y))

Dividing both sides of the equation by 2*19^2, we get:

1 - cos(y) = 144 / (2*19^2)

1 - cos(y) = 144 / 722

1 - cos(y) = 0.199

cos(y) = 1 - 0.199

cos(y) = 0.801

Finding the Value of y
To find the value of y, we can take the inverse cosine (cos^-1) of 0.801. Using a

90 degree.