### Video Transcript

Suppose that the points π΄:
negative three, negative one; π΅: one, two; and πΆ: seven, π¦ form a right-angled
triangle at π΅. What is the value of π¦?

We can go ahead and make a sketch
of these points. π΄ is negative three, negative
one. π΅ is one, two. We know the π₯-coordinate of point
πΆ is seven. And that means πΆ will be located
somewhere along this line. We know the line π΄π΅. And weβve been told that the right
angle of this triangle is at point π΅. We can get a general idea of where
we think point πΆ 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 π΄π΅ is perpendicular to line π΅πΆ. That means the slope of line
segment π΅πΆ is the negative reciprocal of the slope of line segment π΄π΅.

To solve this problem, weβll need
to do three things. First, find the slope of line
segment π΄π΅. Use that slope to find the negative
reciprocal, which is the slope of line segment π΅πΆ. Then, take the slope of line π΅πΆ
and use that to find the π¦-value in point πΆ. If we have two points, we find the
slope by using π equals π¦ two minus π¦ one over π₯ two minus π₯ one. For the points π΄ and π΅, that
would be two minus negative one over one minus negative three, which equals
three-fourths. The slope of line π΄π΅ is then
three-fourths. And weβve 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, weβll take
point π΅: one, two and point πΆ: seven, π¦. Weβll let π΅ be π₯ one, π¦ one and
πΆ be π₯ two, π¦ two. The slope negative four-thirds is
equal to π¦ minus two over seven minus one. Seven minus one is six. To solve this, we cross
multiply. Negative four times six equals
three times π¦ minus two. Negative 24 equals three π¦ minus
six.

To give us a bit more room to solve
for π¦, we add six to both sides and we get negative 18 equals three π¦. Divide both sides of the equation
by three and we get negative six equals π¦. And we found from step three that
π¦ must be equal to negative six. This means, for this to be a right
triangle, point πΆ needs to be located at seven, negative six. And so, we found that missing value
to be negative six.