Case Study Question 03

Mathematics

Chapter 11: Constructions

Class 10

Read the following passage carefully and answer the following questions:

Linking San Francisco with Marin County, the Golden Gate Bridge is a 1.7 mile-long suspension bridge that can be crossed by car, on bicycles or on foot. The Golden Gate Bridge, completed after more than four years of construction, is a landmark recognized almost universally. The Golden Gate Bridge’s 4,200-foot-long main suspension span was a world record that stood for 27 years. The bridge’s two towers rise 746 feet making them 191 feet taller than the Washington Monument. The five-lane bridge crosses Golden Gate Strait.

Question.1.
To divide a line segment AB in the ratio m:n (m, n are positive integers), we draw a ray AX so that `\angle`BAX is an acute angle and then mark points on ray AX at equal distances. What will be the minimum number of these points? 

As we want to divide the given line segment in the ratio m:n, we mark (m + n) points at equal distances on the ray AX drawn as given in the question. This is because we will use Basic proportionality theorem in dividing the line segment in the ratio m : n by joining `A_{m+n}` to `B` and then drawing a line through `A_{m}` parallel to `A_{m+n}B` to intersect AB at C. Then AC : CB = m:n (By BPT). Hence the minimum number of points will be m+n.

Question.2.
Suppose we want to divide the main suspension span of the bridge, which is 4200 feet long, in the ratio 3:4. What will be the minimum number of points which should be marked on ray AX.

As we want to divide the given line segment in the ratio 3:4, we mark 7 (3 + 4) points at equal distances on the ray AX drawn as given in the question. This is because we will use Basic proportionality theorem in dividing the line segment in the ratio 3:4 by joining A to B and then drawing a line through `A_{3}` parallel to AB to intersect AB at C Then AC : CB = 3:4 (By BPT). Hence the minimum number of points to be marked are 7. 

Question.3.
In the above question, points `A_{1}`, `A_{2}`, `A_{3}`, … are located at equal distances on the ray AX drawn such that `\angle`BAX is an acute angle. To which point should point B joined to?

We will use Basic proportionality theorem in dividing the line segment in the ratio 3:4 by joining `A_{7}` to `B` and then drawing a line through `A_{3}` parallel to `A_{3}B` to intersect AB at C. Then AC : CB = 3:4 (By BPT).
Hence, the point joined should be `A_{7}`. 

Question.4.
What is the principal theorem or rule used in the steps to divide the given line segment in any ratio?

Basic Proportionality Theorem.

Question.5.
In the part (B), suppose a ray AX is drawn such that `\angle`BAX is an acute angle. Then a ray BY is drawn parallel to AX and the points `A_{1}`, `A_{2}`, `A_{3}`, … and `B_{1}`, `B_{2}`, `B_{3}`, … are located at equal distances on ray AX and BY, respectively. Then which two points will be joined together? 

`A_{3}` and `B_{4}` will be joined together.

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