Hi there!
I’ve been in danger a lot, mainly when I was an expatriate overseas. Threatened with a knife and almost stabbed in the back, I managed to survive by yelling at the perpetrator, “Go away!” and whatnot.
When I went to Hong Kong, I witnessed a bank robbery just 50 meters ahead of me in a cab, which is still fresh in my memory because the culprit was in an isosceles stance. Scads of police cars were warding off the streets, causing a lot of traffic congestion.
――”Freeze, or I’ll shoot!” a policeman in Cantonese probably said.
Intimidated to be shot, the criminal’s isosceles stance showed his seriousness that he was about to hit the police. I watched the crime scenes and forgot to crouch myself low.
Today, I challenge to find the line segments of an isosceles triangle by taking advantage of the word “isosceles.”
There is the isosceles triangle \(ABC\), whose vertex angle is 36°. Make the point where the bisector of the base angle \(C\) for this triangle intersects the side \(AB\), \(D\). Find the line segments \(DB\) and \(AC\), given \(BC = 1\). (Source: Chart Mathematics Ⅰ＋Ａ Basic Trigonometric Ratio)
Here is my solution to the problem.
As the angle \(\angle ACB = \frac{180° – 36°}{2} = 72°\), \(\angle DCB = \frac{72°}{2} = 36°\).
Regarding the triangles \(\triangle ABC\) and \(\triangle CDB\), \(\angle BAC = \angle DCB = 36°, \angle ACB = \angle CBD = 72°\). Therefore, \(\triangle ABC\text{∽}\triangle CDB\). \(\triangle ABC\text{∽}\triangle CDB\) means that they are homologous triangles to each other.
\(\frac{BC}{AB} = \frac{DB}{CD}\), then \(BC・CD = AB・DB\)・・・（＊）
\(AD = CD = BC = 1\), so put \(DB = x\).
Given \(AB = AD + DB = 1 + x\), （＊）becomes \(1^{2} = (1 + x)x\).
\(x^{2} + x – 1 = 0\), so \(x = \frac{-1\pm\sqrt{5}}{2}\).
\(x = \frac{-1 + \sqrt{5}}{2}\), given \(x>0\).
Therefore, \(DB = \frac{\sqrt{5} – 1}{2}\) and \(AC = AB = 1 + x = \frac{\sqrt{5} + 1}{2}\).
Come to think of it, I had squabbled with my elder brother for his inaccurate division of a piece of cake into two halves. I remember the shape of the cake wasn’t an isosceles triangle.
Stay tuned, and expect to see my next post.
Keep well.
Frank Yoshida
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【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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