In the rectangular coordinate system above, if OP…
We are given a diagram in the coordinate plane. Let’s sketch this below.
We are given that side OP is less than side PQ, and we must determine whether the area of OPQ is greater than 48.
Statement One Alone:
The coordinates of point P are (6,8).
Since we are given the coordinates of point P, we can add a height to triangle OPQ.
We see that we have split triangle OPQ into two right triangles: triangle A and triangle B. We can determine the area of triangle A.
Area of triangle A = (base x height)/2
area = (8 x 6)/2
area = 48/2 = 24
Although it appears that we do not have enough information, we must remember that side OP is less than side PQ. Because PQ is longer than OP, we can determine that the base of triangle B must be longer than the base of triangle A. Triangle A has a base of 6; therefore, triangle B must have a base that is greater than 6. Both triangles have the same height. Therefore, the area of triangle B will be greater than 24, which makes the area of triangle OPQ greater than 48. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.
Statement Two Alone:
The coordinates of point Q are (13,0).
Using the information in statement two we can add in a length of 13 for the base of triangle OPQ.
However, without a height, we cannot determine the area of triangle OPQ. Statement two alone is not sufficient to answer the question.