Daffodil is a beautiful yellow flower
Daffodil is a beautiful yellow flower. Its beauty is celebrated by several counties in Washington State in the USA. In the month of April, several cities hold Daffodil parades simultaneously in their own cities. In Tacoma, the parade is big in which students from 24 high schools participate. The celebrated event also involves a beauty pageant for the selection of the Daffodil Queen. This story in part relates to that joyous festival in which several hundred high school girls enter the pageant as Daffodil princesses with the hope of being singled out for the glory of being selected as the most glamorous and beautiful.
Sara loved Johnny and they frequently went to each other’s place. Most of the time they also went to school and back together. This evening she had phoned him earlier and now she dropped in. Johnny was waiting for her.
Sara: Why do have your travel bag open ?
Johnny’s mom: Yes, we are going for a weekend trip to Tacoma to watch the Daffodil parade.
Sara: Wow, that must be nice. Johnny never mentioned that you guys like daffodils so much.
Johnny’s mom: My brother lives in Tacoma. His daughter, my niece that is, Lizzi has entered into this year’s Daffodil beauty pageant. She is really pretty and might win. She is doing well so far. So, we are going to watch her in the Daffodil parade this weekend. Johnny’s dad was supposed to go but he is busy. So, Johnny and I will go.
As anticipated, they went to Tacoma. Upon returning Johnny had a question for Sara and asked her to come over.
Sara: Hi Johnny. How was the trip ? Did Lizzi win the title ?
Lizzi and Johnny’s argument
Johnny: Lizzi was close. She was a runner up. It was nice to see her and my uncle and aunt, and the parade was beautiful. You know Lizzi is still in grade 11. So she may try it again next year. All that aside but Lizzi and I had an argument. May be you can settle it.
Sara: What kind of argument ?
Johnny: I think it involves geometry.
Sara: Interesting, a beauty queen runner up and geometry ! Sounds like she is smart too. Tell me the exact situation, and I will try to figure it out.
Johnny: Let me draw you this picture of a right angle triangle. We stayed in a hotel which is at C with the roads AC and BC being perpendicular to each other. The parade was going in the direction shown by the arrow. Our question was as to when Lizzi was closest to us during the parade.
I said that it was simple to figure it out. We draw CD from our hotel perpendicular to the parade road AB. Then we can calculate everything.
Sara: You were right.
Johnny: She did not even give me a chance to figure it out and said that she was closest to us when she had crossed one fifth of the parade route between B and A.
Sara: Did she give you a proof ?
Johnny: She said that she would send it me by e-mail but she has not done it yet.
Sara: Why don’t we first see if she was right. Let us say that CD = x, BD = a and DA = b.
BA being the hypotenuse of the right angle triangle ABC, from Pythagoras theorem, AB2 = AC2+ BC2 = 50000 or AB = 223.61 meters.
That means BD + DA = a + b = 223.61 meters.
Now BDC is also a right angle triangle. Therefore DB2 (a2) = BC2 – CD2 (x2). That is a2 = 10000 – x2.
Similarly, from the triangle ADC, b2 = 40000 – x2.
Then b2 – a2 =40000-x2-(10000-x2) = 30000.
That means b2-a2 or (b+a) x (b-a) = 30000
We know that a + b = 223.61 meters
That means b-a =30000/223.61 = 134.16 meters
Adding the last two equations 2b = 357.77 or b = 178.89 meters, and a = 44.7 meters.
Lizzi was right
Johnny: You know, Lizzi was right because a = 44.7 meters and a+b=223.61 meters, and that means a is one fifth of a+b (44.7/223.61).
Sara: Lizzi is smart but how could she figure it out that fast ? One of the possibilities is that she knew what geometric mean is.
What is geometric mean ?
Johnny: What is geometric mean ?
Sara: Commonly we talk about average which is arithmetic mean in which you add two numbers and the divide the sum by two. In geometric mean you multiply them and take the square root of the product (see Trip To Langley Park to read more about geometric mean).
Johnny: I see it is something like saying that the area of a rectangle with the sides a and b will be the same as that of a square with all sides being √ab. But how does that fit in here ?
Sara: The picture you drew shows a baseline AB divided into two parts BD (a) and DA (B). In geometry when you draw a right angle triangle ABC, the vertical line from C onto D is the geometric mean.
Johnny: Does that mean x = √ab and x2 = ab ?
Sara: You can start from there and from a + b = 223.61meters. Now in triangle BDC, x2 =1002 -a2.
Johnny: then ab = 1002 – a2 or b = 10000/a – a or a + b = 10000/a or 223.61 =10000/a or a = 10000/231.61 = 44.7 meters. Of course, then a/(a+b) = 44.7/223.61 = 1/5. That’s clever. It still involves a lot of calculations. Also, what is the proof that x2 = ab ? We just assumed that to be the definition.
Sara: You can see that the ∠ADC = ∠ACB = 90°.
∠DAC = 90° -∠DCA because sum all angles in a triangle is 180°.
Also ∠BCD = 90° -∠DCA because ACB is a right angle.
This makes ∠DAC =∠BCD
Therefore being the third angles ∠ACD =∠DBC.
Therefore, triangles ACD and BCD are similar.
Johnny: Therefore, BD/CD = CD/AD or b/x = x/a or x2 = ab. That’s neat.
Sara: Lizzi is a smart girl. What does she plan on doing after high school ?
Johnny: She wants to become a civil engineer but I don’t know what her parents want her to do.
Sara: I am impressed with her math skills. Hope she gets to do whatever she likes.
Challenge
The sides of the right angle triangle ABC are AB = a and BC =b. A square with an arm of p is inscribed in it as shown. How is p related to a and b?
Solution:
Areas of different shapes appearing in the figure are
Triangles: ABC = ab/2, ADE = (a-p) x p/2, EFC = (b-p) x p/2, Square BDEF = p2.
Triangle ABC is the sum of trianges ADE, EFC and the square.
Therefore, ab/2 = (a-p) x p/2 + (b-p) x p/2 + p2 or
ab/2 = (a + b) x p/2 – p2/2 – p2/2 + p2 or
ab/2 = (a+b) x p/2 or
p/2 = ab/(2(a+b)) or
p = ab/(a+b) or p = 1/(1/a + 1/b)
Harmonic mean of a and b would be 2/(1/a + 1/b)
Therefore p = half of Harmonic mean of a and b
(see Trip To Langley Park to read more about harmonic mean)
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