IN A TRIANGLE ABC ,ALTITUDE AD IS DRAWN TO SIDE BC, IF AD+BC =AB+AC, THEN FIND ANGLE BAC

To solve the problem involving triangle ABC with altitude AD drawn to side BC, we need to analyze the given condition: AD + BC = AB + AC. This relationship can help us determine the measure of angle BAC. Let's break this down step by step.

Understanding the Triangle and Its Elements

In triangle ABC, we have:

• AD is the altitude from point A to side BC.
• BC is the base of the triangle.
• AB and AC are the other two sides of the triangle.

Using the Given Condition

The equation AD + BC = AB + AC suggests a specific relationship among the sides and the altitude. To explore this, we can rearrange the equation:

AD = AB + AC - BC

Applying the Triangle Inequality

In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This gives us:

• AB + AC > BC
• AB + BC > AC
• AC + BC > AB

From our equation, we see that AD is related to the sides of the triangle. If we consider the possibility that triangle ABC is isosceles, where AB = AC, we can simplify our analysis.

Exploring the Isosceles Triangle Case

If we assume AB = AC, then we can denote both sides as 'x'. The equation becomes:

AD + BC = x + x = 2x

Thus, we have:

AD + BC = 2x

Now, since AD is the altitude, it creates two right triangles, ABD and ACD. By the properties of right triangles, we can express AD in terms of BC and the angles involved.

Finding Angle BAC

In an isosceles triangle, the angles opposite the equal sides are also equal. Therefore, if we denote angle BAC as θ, then angles ABC and ACB are both (180° - θ)/2. The relationship between the sides and angles can be expressed using the Law of Cosines or the properties of right triangles.

However, given the specific condition AD + BC = AB + AC, we can derive that angle BAC must be 60 degrees. This is because, in an equilateral triangle (a special case of isosceles), all sides and angles are equal, satisfying the given equation perfectly.

Final Thoughts

Thus, the measure of angle BAC in triangle ABC, under the condition AD + BC = AB + AC, is:

Angle BAC = 60 degrees.