In the figure, the coordinates of points 𝐴, 𝐵, and 𝐶 are six, three; eight, three; and six, seven, respectively. Determine the lengths of the line segments 𝐴𝐶 and 𝐴𝐵. Then calculate the area of triangle 𝐴𝐵𝐶, where a length unit is equal to one centimetre.
Let’s begin by adding the coordinates we’ve been given for points 𝐴, 𝐵, and 𝐶 onto our diagram. 𝐴 is six, three. 𝐵 is eight, three. And 𝐶 is six, seven. The question asks us to determine the lengths of the sides 𝐴𝐶 and 𝐴𝐵.
Now, 𝐴𝐵, first of all, is a horizontal line. We can see this because both of its endpoints have the same 𝑦-coordinate of three. The length of this line then will just be the difference in the 𝑥-coordinates of its two endpoints, eight minus six, which is equal to two. We’re told that a length unit is one centimetre. So the units for this length are two centimetres.
For the line 𝐴𝐶, we can see that this is a vertical line because this time both of its endpoints have the same 𝑥-coordinate. They both have an 𝑥-coordinate of six. The length of this line then is the difference in the 𝑦-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 𝐴𝐵 is horizontal and the side 𝐴𝐶 is vertical, there is a right angle between the sides 𝐴𝐵 and 𝐴𝐶. 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 𝐴𝐵. And the height will be the length of 𝐴𝐶, both of which we’ve just calculated.
So the area of triangle 𝐴𝐵𝐶 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 𝐴𝐶 is four centimetres. The length of the side 𝐴𝐵 is two centimetres. And the area of triangle 𝐴𝐵𝐶 is four centimetres squared.