Answer:
Explanation:
Given: In triangle ABC, AB = AC and D is the midpoint of BC.
To prove:
i) AD is perpendicular to BC.
We can prove this by using the midpoint theorem, which states that a line segment connecting the midpoints of two sides of a triangle is perpendicular to the third side and half as long as the third side.
Since D is the midpoint of BC, we have BD = DC. Thus, the line segment AD connects the midpoints of sides AB and AC of triangle ABC.
By the midpoint theorem, AD is perpendicular to BC and AD = BC / 2.
ii) AD bisects angle BAC
We can prove this by using the fact that if two line segments are perpendicular and one is an altitude of a triangle, then the two line segments divide the triangle into four smaller triangles that are congruent by SAS congruence.
Since AD is perpendicular to BC, we have two right triangles, ABD and ACD. By the Pythagorean theorem, we have AB = AC, which means that ABD and ACD are congruent by SAS congruence.
Since ABD and ACD are congruent, we have angle BAD = angle CAD. Also, since angle BAC is the sum of angles BAD and CAD, we have angle BAC = 2 * angle BAD.
Therefore, by substitution, we have angle BAC = 2 * angle CAD. This means that angle CAD is half of angle BAC, or in other words, AD bisects angle BAC.
Hence, we have proved that in triangle ABC, AD is perpendicular to BC and AD bisects angle BAC.