Video Transcript
Find the length of the perpendicular line drawn from the point 𝐴: negative eight, five to the straight line that passes through the point 𝐵: two, negative four and whose slope is equal to negative eight.
So to solve this problem, we, in fact, have a formula that we can use, and this helps us find the length of the perpendicular line drawn from a point to a straight line. However, we need to have our point, which is 𝑥 one, 𝑦 one. And we have to have our straight line in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero. Well, what this formula is is that 𝐿, and that’s our perpendicular distance, is equal to the modulus or absolute value of 𝑎𝑥 sub one plus 𝑏𝑦 sub one plus 𝑐. And this is all over the square root of 𝑎 squared plus 𝑏 squared.
Well, this is great because we have a formula we can use. However, we don’t yet have our straight line in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero. So that’s gonna be the first thing we’re gonna work out. But how are we gonna do that? Well, we’re gonna do it using the general form for the equation of a straight line. And that is that 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 is the slope and 𝑐 is the 𝑦-intercept. Well to start us off, what we have is 𝑦 is equal to negative eight 𝑥 plus 𝑐. And that’s because the slope is equal to negative eight. But we don’t know 𝑐, so how can we find 𝑐?
Well, we know a point on a straight line cause we know that the straight line passes through the point 𝐵, which is two, negative four. So therefore, we can substitute these in for our 𝑥- and 𝑦-values. So 𝑥 is equal to two, and 𝑦 is equal to negative four. And we can use this to find out what 𝑐 will be. So therefore, when we do this, what we’re gonna get is negative four equals negative eight multiplied by two plus 𝑐. So we’re gonna get negative four is equal to negative 16 plus 𝑐. So then what we can do to find 𝑐 is add 16 to each side of the equation. And when we do that, what we’re gonna get is 12 is equal to 𝑐. So now we’ve got the equation of our straight line cause we can substitute this in. So when we do that, we’re gonna get 𝑦 is equal to negative eight 𝑥 plus 12.
However, this isn’t quite in the form that we wanted because we want it in the form 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero. So, we need to do a bit of rearranging to do that. So if we add eight 𝑥 and subtract 12 from each side of the equation, what we’re gonna get is eight 𝑥 plus 𝑦 minus 12 equals zero. So now we have it in the form that we wanted. And we have our 𝑎-, 𝑏-, and 𝑐-values; so 𝑎 is equal to eight, 𝑏 is equal to one, and 𝑐 is equal to negative 12. So, as we said, we’ve got 𝑎, 𝑏, and 𝑐. However, to use our formula, what we also need is our 𝑥 sub one, 𝑦 sub one. Well, our 𝑥 sub one and 𝑦 sub one are going to be the coordinates of the point from which our perpendicular line is drawn. So we’re gonna have 𝑥 sub one is equal to negative eight and 𝑦 sub one is equal to five.
So now we have all our constituent parts. We can put them into our formula to find the length of our perpendicular line. So therefore, we can say 𝐿 is equal to the modulus or absolute value of eight multiplied by negative eight plus one multiplied by five minus 12. And this is all over root eight squared plus one squared. So therefore, we’re gonna have 𝐿 is equal to the absolute value or modulus of negative 64 plus five minus 12 over root 65. So therefore, 𝐿 is equal to the modulus or absolute value of negative 71 over root 65, which is just equal to 71 over root 65 because if it’s the absolute value or the modulus, then therefore we’re only interested in the magnitude or positive value.
Well now, we could just have the value as 71 over root 65. However, if we’ve got a square root or surd on the bottom, on the denominator, then what we want to do is rationalize the denominator. And to do that, we multiply by root 65 over root 65. But why do we do that? Well, we do it because if we have root 𝑎 multiplied by root 𝑎, this is just equal to 𝑎. So therefore, if we multiply root 65 by root 65, we’re just gonna get 65. So therefore, we can say that the length of the perpendicular line drawn from the point 𝐴: negative eight, five to the straight line that passes through the point 𝐵: two, negative four and whose slope is equal to negative eight is 71 root 65 over 65.
