1) The document investigates how the width of an alleyway does not affect the height of intersection between two ladders when their reaching heights remain the same. 2) Using Pythagorean theorem and linear equations, the author models the ladders and finds the intersection points for alley widths of 250cm, 300cm, 350cm, and 400cm. 3) It is shown that in all cases the height of intersection is 120cm from the ground, demonstrating that the alley width does not impact this measurement when the ladders' reaching heights are held constant.

1. Julia Li
Ladders
This Investigation is to investigate why the width of alleyway doesn’t affect the height of
intersection point of 2 ladders when the height that the ladders reach stays the same.
The length of the alleyway is 250cm, ladder a reaches 280cm, ladder b
reaches 210cm.
a
b
First I shall give a pre-exercise about the ladders.
Finding the length of the ladders.
The length of the ladders affects the height that the ladder reaches.
When the ladder leans on the side of the wall, they form a right-angled
triangle. Since the length of the ladder is the longest side of the triangle
I shall use the Pythagoras Theorem formula to solve the length of the
ladder.
(Pythagoras Theorem Formula)
Ladder a: Ladder b:
375.37cm 325.58cm
Therefore, I used the Pythagoras Theorem to find the length of the ladder.
The length seems a bit inaccurate...so if imagining the x and y axes are shown on the diagram...the
next step shall be...
Making equations for ladders.
I used the linear equation (y=mx+c) to write the expressions of ladders.
Ladder a: Ladder b:
As you can see I multiplied every
number by 100 at 4th step because, so I
can convert the decimal points into
whole numbers.

2. Julia Li
Drawing the Line Graph.
*Find the yellow line on Figure 1. It represents the line when the alleyway is 250cm wide.
Intersection point: (142.8572, 120) 00
The intersection point is 120cm away from the ground.
Because our investigation is investigating how the width doesn’t affect the height between the
intersection point and ground. Therefore I’m going to find...
Different Widths.
I’ll add another 3 widths, that will prove it. (300cm, 350cm, 400cm)
If the width is 300cm...
Ladder a: Ladder b:
Key: Red Line
Intersection Point:
(171.4286, 120)
If the width is 350cm...
Ladder a: Ladder b:
Key: Blue Line
Intersection Point:
(200, 120)
If the width is 400cm...
Ladder a: Ladder b:
Key: Green Line
Intersection Point:
(228.5714, 120)
Now I’m going to draw it on the graph, so I can see the pattern.

3. Julia Li
(Figure 1)
Key: 250cm width
alleyway is the Yellow
Line
Equations
Since I got 4 examples, I will give one linear equation for each ladder that qualifies for all lines. By
substituting w as the width.
Ladder a:
Original Equation:
Since 250 is the width, I simply substituted w to 250.
Therefore:
Ladder b:
Original Equation:
Since 250 is the width, I simply substituted w to 250 again.
Therefore:
But why is the height always the same?
Explanation of Height of Intersection
By logic, if the length of the ladder is able to be changed, the width doesn’t change the height
between intersection point and ground. Since the height that the ladder reaches is the same, and
both of them stays the same, the width doesn’t affect the height between intersection point and
ground.
But to give a better explanation, I will show you how the weight by extrapolating the formulas.

4. Julia Li
The intersection point of the two lines has to conform both of the formulas.
If the intersection point =
I can find .
To find the , simply use both of the formulas. Because the intersection point has to conform both
of the formulas.
Since I found the value of . Then I can find the value of .
To find the value of , I substitute into one another formula again. I chose to do the one for
Ladder b.
When I extrapolate the equation, you can see that the answer is 120 and the w is crossed out.
Which proves us that the width doesn’t matter with the height of intersection.
Now lets make a more universal equation, an equation that can be used for any height of ladders.
Final Equation.
Ladder a: the one that reaches 280cm
Ladder b: the one that reaches 210cm
Therefore a=280, b=210 for substitution
Then I shall substitute it from the equation I made previously.

5. Julia Li
Therefore the formula is:
This formula can be used for any height that the ladder reaches.
a & b are the height that the two ladders reaches. The formula is to find the height of the
intersection point.