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In a /\ ABC, AB = AC and AD bot BC, then...

In a `/_\ ABC`, AB = AC and `AD bot BC`, then BD = .......

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To solve the problem, we need to find the length of segment BD in triangle ABC where AB = AC and AD is perpendicular to BC. ### Step-by-Step Solution: 1. **Identify the Given Information**: - Triangle ABC is isosceles with AB = AC. - AD is the altitude from A to BC, meaning AD is perpendicular to BC. 2. **Draw the Triangle**: - Draw triangle ABC with points A, B, and C. - Mark AB = AC to indicate that it is isosceles. - Draw a perpendicular line from A to BC, marking the foot of the altitude as point D. 3. **Label the Angles**: - Since AB = AC, by the properties of isosceles triangles, the angles opposite to these sides are equal. Therefore, angle B = angle C. 4. **Consider Triangles ABD and ACD**: - Now, look at triangles ABD and ACD. - We know: - AB = AC (given) - AD = AD (common side) - Angle B = Angle C (since AB = AC) 5. **Use Congruence**: - By the Side-Angle-Side (SAS) postulate, triangles ABD and ACD are congruent. - Therefore, by Corresponding Parts of Congruent Triangles (CPCT), we have BD = DC. 6. **Relate BD and BC**: - Since BD = DC, we can say that BD is half of BC. - Therefore, BD = 1/2 * BC. ### Final Answer: BD = 1/2 * BC.
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