What kind of shape is quadrilateral ? It looks like it might be a rhombus. To check, we can calculate the length of each side. Using the Pythagorean Theorem, we find that the lengths of segments and are units, and the lengths of segments and are units. All side lengths are between 6 and 7 units long, but they are not exactly the same. So our calculations show that is not really a rhombus, even though at first glance we might think it is.

We did just show that two pairs of opposite sides of are congruent. This means that must be a parallelogram. Checking slopes confirms this. Sides and both have a slope of . Sides and both have a slope of 6.

Can we find the area of triangle ? That seems tricky, because we don’t know the height of the triangle using as the base. However, angle seems like it could be a right angle. In that case, we could use sides and as the base and height.

To see if is a right angle, we can calculate slopes. The slope of is or , and the slope of is . Since the slopes are opposite reciprocals, the segments are perpendicular and angle is indeed a right angle. This means that we can think of as the base and as the height. The length of is 10 units and the length of is 5 units. So the area of triangle is 25 square units because .