Find the distance between the points (6,1) and (9,5)
2 answers:
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Answer:
AB = 30 inch
BC = 14 inch or BC = 50 inch
Step-by-step explanation:
Make a drawing. Please see the attachment. All measurements are in inches.
Consider the two triangles:
∆ ABC and ∆ AB'C
1. Looking only at ∆ ABC. Since this is <em>not</em> a rectangular triangle, let's first consider ∆ ANC, so we can calculate NC.
AN is the height and there fore is perpendicular to NC and thus also to BC.
In ∆ ANC
AN² + NC² = AC²
We want to calculate NC, so:
NC² = AC² - AN²
Given: AC = 40 and AN = 24
NC² = 40² + 24²
NC² = 1600 + 576
NC² = 1024
NC = +-SQRT(1024)
NC = 32
NC = NB + BC
We want to know BC
BC = NC - NB
Given: NB = BN = 18 and we just calculated NC to be 32 so...
BC = 32 - 18
BC = 14
2. Looking only at ∆ AB'C
AN is the height and there fore is perpendicular to B'C. Let's consider ∆ AB'N, so we can calculate AB'.
AN is the height and there fore is perpendicular to B'N, which means it has an angle of 90° in ∆ AB'N.
In ∆ AB'N
c² = a² + b²
AB'² = AN² + B'N²
Given: AN = 24 and B'N = 18
AB'² = 24² + 18²
AB'² = 576 + 324
AB'² = 900
AB' = +-SQRT(900)
AB' = 30
In ∆ AB'C
with AB' = 30 and AC = 40
c² = a² + b²
B'C² = AB'² + AC²
B'C² = 30² + 40²
B'C² = 900 + 1600
B'C² = 2500
B'C = +-SQRT(2500)
B'C = 50
Now we have our answers.
Extra:
Please look at the picture again, but now concentrate on the indicated 50 and - 50...
I am trying to explain something about the meaning of the outcome of mathematical calculations like:
c² = 2500
c = +- SQRT(2500)
c = 50 or c = - 50
Depending on where you want to start from, you can "move" 50 inch in one direction or 50 inch in the opposite direction, hence the -50 inch. Please let me explain why I am making a fuss...
1). Let's consider ∆ AB'C. Going from B' to C, you "move" -50 inch form B' towards C. This 50 inch is what we calculated earlier, but the we neglected to explain why we discarded the minus value of the SQRT... We just stated it to be only the positive value! Normally we give no meaning to the negative variant of it... Well, is this true in this case?
2). Let's consider ∆ ABC'. Going from B to C', you "move" 50 inch form B towards C'. Please understand that the direction is opposite that of - 50 inch in 1).
It is important to understand that a negative sign means the 180° in the direction of the other way.
In general. When calculating a square side by using the SQRT, you carefully need to consider if you can discard the -
minus value of your calculated outcome.
Not always, but <em>usually</em> there is some sort of meaning to the negative part of the SQRT, you just need to be willing to understand what it possibly could mean.
I hope this has made some sense to you :-).
