Video Transcript
Suppose that the points 𝐴:
negative three, negative one; 𝐵: one, two; and 𝐶: seven, 𝑦 form a right-angled
triangle at 𝐵. What is the value of 𝑦?
We can go ahead and make a sketch
of these points. 𝐴 is negative three, negative
one. 𝐵 is one, two. We know the 𝑥-coordinate of point
𝐶 is seven. And that means 𝐶 will be located
somewhere along this line. We know the line 𝐴𝐵. And we’ve been told that the right
angle of this triangle is at point 𝐵. We can get a general idea of where
we think point 𝐶 would be. But this is not a good way to find
an accurate answer. But because we know this is a right
triangle, we could say that line 𝐴𝐵 is perpendicular to line 𝐵𝐶. That means the slope of line
segment 𝐵𝐶 is the negative reciprocal of the slope of line segment 𝐴𝐵.
To solve this problem, we’ll need
to do three things. First, find the slope of line
segment 𝐴𝐵. Use that slope to find the negative
reciprocal, which is the slope of line segment 𝐵𝐶. Then, take the slope of line 𝐵𝐶
and use that to find the 𝑦-value in point 𝐶. If we have two points, we find the
slope by using 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. For the points 𝐴 and 𝐵, that
would be two minus negative one over one minus negative three, which equals
three-fourths. The slope of line 𝐴𝐵 is then
three-fourths. And we’ve completed step one.
For step two, we need to take the
negative reciprocal of the slope we found in step one. The negative reciprocal of
three-fourths is negative four-thirds. And that is step two. Now, for step three, we’ll take
point 𝐵: one, two and point 𝐶: seven, 𝑦. We’ll let 𝐵 be 𝑥 one, 𝑦 one and
𝐶 be 𝑥 two, 𝑦 two. The slope negative four-thirds is
equal to 𝑦 minus two over seven minus one. Seven minus one is six. To solve this, we cross
multiply. Negative four times six equals
three times 𝑦 minus two. Negative 24 equals three 𝑦 minus
six.
To give us a bit more room to solve
for 𝑦, we add six to both sides and we get negative 18 equals three 𝑦. Divide both sides of the equation
by three and we get negative six equals 𝑦. And we found from step three that
𝑦 must be equal to negative six. This means, for this to be a right
triangle, point 𝐶 needs to be located at seven, negative six. And so, we found that missing value
to be negative six.
