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The shortest distance and the y-intercept

Autor:Callum Marshall
In this investigation you will explore how the distance from a point to a line changes as the y-intercept of the line changes. Note: the shortest distance from a point to a line is the perpendicular distance. AD in the diagram below. Also, it will be easier for you to see patterns if at least the decimal results for the change in distance you find are written as fractions where possible. 

TASK

The diagram below shows a line with equation, . Sliders have been set up so that the values of the parameters can be changed. Also, a point A has been placed on the diagram.

Section A; Level 1 - 4

1. For the line shown above,DESCRIBE how the value of the y-intercept is related to the values of b and c? 2. To check your understanding of the mathematics involved in this investigation, set a = 6, b = 8, c = -32 and point A as (4, 5). Then, using an ALGEBRAIC method verify that the length of AB is 3.2. The steps involved are;
  • find the equation of the line AB
  • find where AB intersects the line in order to find the coordinates of point B
  • use the distance formula to find the length of AB
Note: the distance formula is: From now on you will use the diagram above to find the information you need. 3. Set a = -3 and b = 4. i. Set the point A to (4, 1) and then find the distance AB when the y-intercept is 1, 2, 3, etc. Record the y-intercept, distance and change in distance data in a table. ii. Describe how the distance AB changes as the y-intercept increases - write the change in distance as a fraction in simplest form. ii. Change the position of point A to (5, 3) and again, find the distance AB when the y-intercept is 1, 2, 3, etc. iv. How does changing the position of point A impact the pattern you described in part ii. above? 4. Set a = -7 and b = 24 i. Set the point A to (4, 0) and again, find the distance AB when the y-intercept is 1, 2, 3, etc. Record the distance and y-intercept data in another table. ii. Describe any patterns in the way the distance changes - write the change in distance as a fraction in simplest form. 5. Set a = -5 and b = 12 i. Keep the point A at (4, 0) and again, find the distance AB when the y-intercept is 1, 2, 3, etc. Record the distance and y-intercept data in another table. Note: Calculate the change in distance correct to at least 7-decimal places. And, use the decimal to fraction convertor shown below to change to a fraction (you will have to check the approximate box). ii. Describe any patterns in the way the distance changes this time.

Decimal to fraction convertor - scroll down and use the approximate option.

Decimal to fraction convertor - scroll down and use the approximate option.

Decimal to fraction convertor - scroll down and use the approximate option.

https://www.mathsisfun.com/converting-decimals-fractions-solver.html
6. By considering the data you have collected so far, and any patterns you have observed, what conjecture can you make about the relationship between the CHANGE in the shortest distance from a point to a line as the y-intercept changes and the values of the parameters a and b when

Write a RULE that fits your conjecture.

7. Use your rule to predict the relationship between the CHANGE in distance to the line and the y-intercept when; a = -9 and b = 40 ? 8. Test your prediction from question 7, using the diagram above.

Section B; Level 5 - 8

9. By selecting other values for a and b, explore the situation further. To help you do this try some of the following suggestions: i. look at the examples you have already used and switch the values for a and b around.  And, you could also look at what happens when you change the signs of a and b. ii. try setting a = 8 and b = -15, or a = -15 and b = 8, etc iii. try doubling some of the values you have already used for a and b. So where a = -3 and b = 4, try a = -6 and b = 8, etc iv. try setting a = 48 and b = -55, etc In this section you MUST show evidence that you have;
  • Selected and applied problem solving techniques to discover PATTERNS in the new data you have collected,
  • used the new data you collected to decide whether you need to change your previous RULE from Question 6 to write a more general rule,
  • VERIFY your general rule

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