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Three lines are determined by three points unless the points are all on the same line ( i.e. co-linear)

Q: How many lines are determined by A B and C?

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4*3/2 = 6 lines.

It depends very much on what a, b and c are.

No. In plane geometry, two lines can intersect at at most one point. This means that for n points, the maximum number of intersections is limited by the number of pairs of lines. For lines a,b,c,d, there are six pairs: (a,b), (a,c), (a,d), (b,c), (b,d), (c,d). So four lines can have at most six intersections.

y = mx + c y = mx + b We see that the lines have the same slope m. If the lines have slopes with the same sign, then the lines are parallel. Soppose that the slope is positive, m > 0. Case 1: If c > 0, b> 0, and c > b, or c < 0, b < 0, and c < b, then we have to work in the same way to find the distance between lines, which is also the same distance. Let work for the first possibility where c > 0, b > 0, and c > b. If we draw the lines, we see that from the interception of the lines with x and y-coordinate axis, are formed two similar right triangles on the second quadrant. The bigger triangle vertices are (0, c), (0, 0), and (-c, 0) The midpoint of the hypotenuse is (-c/2, c/2). The distance between the origin and the midpoint is (c√2)/2. √[(-c/2)2 + (c/2)2] = √(c2/4 + c2/4) = √ (2c2/4) = (c√2)/2 The smaller triangle vertices are (0, b), (0, 0), and (-b, 0) The midpoint of the hypotenuse is (-b/2, b/2). The distance between the origin and the midpoint is (b√2)/2. √[(-b/2)2 + (b/2)2] = √(b2/4 + b2/4) = √ (2b2/4) = (b√2)/2 The distance between the lines is (√2)/2)(c - b). (c√2)/2 - (b√2)/2 = (√2)/2)(c - b) Case 2: If c > 0, b< 0 or c < 0, b > 0. If we draw the lines, we see that from the interception of the lines with x and y-coordinate axis are formed two similar right triangles on the second and fourth quadrant. In this case the distance between lines is (√2)/2)(c + b). If the slope is negative for both lines, then we find the same result as above, but the formed right triangles are in the first and third quadrant. If the lines have slopes with different sign, then the lines intersect.

10 lines, but only if no three of them are collinear.

Related questions

# 1

5 its 4

4*3/2 = 6 lines.

exactly one and only one.

If points A, B, and C are not on the same line, they determine a single plane.

It depends very much on what a, b and c are.

The rhyme scheme of "Madam and the Rent Man" is AABB. This means that the first and second lines rhyme with each other, and the third and fourth lines rhyme with each other.

There are 5 RER lines in Paris: A, B, C, D, and E.

b & c

exactly one

B and C

No. In plane geometry, two lines can intersect at at most one point. This means that for n points, the maximum number of intersections is limited by the number of pairs of lines. For lines a,b,c,d, there are six pairs: (a,b), (a,c), (a,d), (b,c), (b,d), (c,d). So four lines can have at most six intersections.