Analytic geometry studies geometric objects using coordinate systems and algebraic equations, translating shapes into formulas so that algebra can solve geometry problems. This field, founded by Descartes, unifies algebra and geometry.
It translates shapes into equations so algebra can solve geometry problems.
Showing a random 20 of 50 problems.
Example 1
medium
The equation y=x2 represents what shape?
Example 2
challenge
Find the distance from the point (0,0) to the line 3x+4y−10=0 using the distance formula for a point to a line.
Example 3
medium
Does the point (3,4) lie on the circle x2+y2=25?
Example 4
hard
Find the x-intercept of the line 5x−2y=20.
Example 5
medium
The vertices of a triangle are A(0,0), B(6,0), C(0,8). Find its area.Area of triangle A(0,0), B(6,0), C(0,8) = ½ × 6 × 8 = 24.
Example 6
easy
Find the midpoint of A(2,4) and B(8,10).
Example 7
medium
Find the slope of a line perpendicular to y=32x+5.
Example 8
challenge
Find the intersection of the lines y=2x+1 and y=−x+7.
Example 9
hard
Verify that the quadrilateral with vertices A(0,0), B(4,0), C(4,3), D(0,3) is a rectangle and find its diagonal.Rectangle with sides 4 and 3: diagonal = √(16+9) = 5.
Example 10
easy
The equation y=mx+b represents what, and what do m and b mean?
Example 11
easy
What is the radius of the circle x2+y2=36?
Example 12
easy
Who is credited with founding analytic geometry?
Example 13
medium
Find the midpoint of segment AB where A(−3,7) and B(5,−1).Find the midpoint M of segment AB.
Example 14
easy
Find the slope of the line through (1,3) and (5,11).
Example 15
easy
Write the equation of a line with slope 4 and y-intercept −3.
Example 16
medium
Find the y-intercept of the line 2x+3y=12.
Example 17
challenge
Show that the points (1,1), (4,1), (4,5) form a right triangle using slopes.Right angle at B(4,1): horizontal leg AB ⊥ vertical leg BC.
Example 18
easy
Determine whether points P(1,2), Q(3,6), and R(5,10) are collinear using the slope method.Are P, Q, R collinear? Compare slopes PQ and QR.
Example 19
hard
Find the equation of the line that passes through (4,−1) and is parallel to 3x+2y=8.