### Video Transcript

In the figure, the coordinates of points π΄, π΅, and πΆ are six, three; eight, three; and six, seven, respectively. Determine the lengths of the line segments π΄πΆ and π΄π΅. Then calculate the area of triangle π΄π΅πΆ, where a length unit is equal to one centimetre.

Letβs begin by adding the coordinates weβve been given for points π΄, π΅, and πΆ onto our diagram. π΄ is six, three. π΅ is eight, three. And πΆ is six, seven. The question asks us to determine the lengths of the sides π΄πΆ and π΄π΅.

Now, π΄π΅, first of all, is a horizontal line. We can see this because both of its endpoints have the same π¦-coordinate of three. The length of this line then will just be the difference in the π₯-coordinates of its two endpoints, eight minus six, which is equal to two. Weβre told that a length unit is one centimetre. So the units for this length are two centimetres.

For the line π΄πΆ, we can see that this is a vertical line because this time both of its endpoints have the same π₯-coordinate. They both have an π₯-coordinate of six. The length of this line then is the difference in the π¦-coordinates of the endpoints, seven minus three, which is equal to four. And again, the units for this length will be centimetres.

If the side π΄π΅ is horizontal and the side π΄πΆ is vertical, there is a right angle between the sides π΄π΅ and π΄πΆ. So we have a right triangle. To find the area of a triangle, we use the formula a half multiplied by base multiplied by perpendicular height. So in our right triangle, the base will be the length of π΄π΅. And the height will be the length of π΄πΆ, both of which weβve just calculated.

So the area of triangle π΄π΅πΆ will be given by a half multiplied by two multiplied by four. A half multiplied by two or a half of two is just one. So our calculation simplifies to one multiplied by four. This is just equal to four. And the units for this area will be centimetres squared because the length units were centimetres.

So we found that the length of the side π΄πΆ is four centimetres. The length of the side π΄π΅ is two centimetres. And the area of triangle π΄π΅πΆ is four centimetres squared.