Problem:
Two lines with slopes and 2 intersect at . What is the area of the triangle enclosed by these two lines and the line
Answer Choices:
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Solution:
Let's first work out the slope-intercept form of all three lines: and implies so , while implies so . Also, implies . Thus the lines are , and . Now we find the intersection points between each of the lines with , which are and . Using the distance formula and then the Pythagorean Theorem, we see that we have an isosceles triangle with base and height , whose area is .
OR
Like in other solutions, we find that the three points of intersection are , and . By the Pythagorean theorem, this is an isosceles triangle with base and equal length . The area of an isosceles triangle with base and equal length is . Plugging in and ,
The problems on this page are the property of the MAA's American Mathematics Competitions