Waste less time on Facebook — follow Brilliant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
2D Coordinate Geometry
In the 1600s, René Descartes married algebra and geometry to create the Cartesian plane.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Let \(P\) be the internal division point of the line segment \(AB\) joining points \(A=(-9,6)\) and \(B=(12,4)\) in the ratio \(t:(1-t)\) for \(t>0\). If \(P\) lies in the first quadrant of the Cartesian plane, what is the range of the real number \(t?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Consider the line segment joining points \(A=(-6,-7)\) and \(B=(8,-1)\). If \(P\) is the intersection point of this line segment and the \(y\text{-axis},\) and \(P\) internally divides the line segment in the ratio \(m:n,\) where \(m\) and \(n\) are coprime positive integers, what is the value of \(mn?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Let \(P\) be the internal division point of \(\overline{AB}\) joining points \(A\) and \(B\) in the ratio \(7:8,\) and \(Q\) be the external division point of \(\overline{AB}\) in the ratio \(8:7.\) If the length of \(\overline{PQ}\) is \(\lvert\overline{PQ}\rvert=113,\) what is \(\lvert\overline{AB}\rvert?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Let \(R\) be the external division point of the line segment \(PQ\) joining points \(P=(-1,1)\) and \(Q=(11,16)\) in the ratio \(k:5.\) If \(P\) is on the line \(x+y=-18,\) what is the positive number \(k?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
The line segment \(AB\) connecting two points \(A=(23, 14)\) and \(B=(2, 5)\) on the xy-plane is internally divided \(2:1\) by a point \(P=(x, y\)). What is \(x+y\)?