HMMT 二月 2021 · 几何 · 第 3 题
HMMT February 2021 — Geometry — Problem 3
题目详情
- Triangle ABC has a right angle at C , and D is the foot of the altitude from C to AB . Points L , M , and N are the midpoints of segments AD , DC , and CA , respectively. If CL = 7 and BM = 12, 2 compute BN .
解析
- Triangle ABC has a right angle at C , and D is the foot of the altitude from C to AB . Points L , M , and N are the midpoints of segments AD , DC , and CA , respectively. If CL = 7 and BM = 12, 2 compute BN . Proposed by: Hahn Lheem Answer: 193 Solution: Note that CL , BM , and BN are corresponding segments in the similar triangles 4 ACD ∼ 4 CBD ∼ 4 ABC . So, we have CL : BM : BN = AD : CD : AC. 2 2 2 2 2 2 Since AD + CD = AC , we also have CL + BM = BN , giving an answer of 49 + 144 = 193.