The points A and B have coordinates (-2, 1) and (5, 2) respectively.

Question

Find, in its simplest surd form, the length AB

Find an equation of the line through A and B, giving your answer in the form ax+by+c=0, where a, b and c are integers.

The line through A and B meets the x-axis at the point C. Find the coordinates of C.

Answer 1

When you have the coordinates of two points you can use the Pythagorean theorem to find the distance between the two points. Use the point coordinates to find the change of the y value and the corresponding change of the x value. The points are (-2,1) and (5,2), so the difference between the y values is 7 and the difference between the x values is 1. These are the lengths of the two legs of a right triangle where the hypotenuse is the length of the segment AB. So using a² + b² = c^{2 ,}we get 7² + 1² = c², or 49 = 1 = c². So c² is equal to 50, and c is equal to √50. That needs to be simplified, and can be split to √25 x √2, which becomes 5√2 for the length of segment AB.

For the equation of the line, use the point slope formula, y - y₁ = m(x - m1). The slope is the chage of y divided by the change of x, and we can get that from the first part of the problem. Y changed by 7 units, and x changed by 1, so the slope is 7. Use the point (-2,1) to plug into the formula because it has simple numbers. So, y - (-2) = 7 (x - 1). Simplify that to y + 4 = 7x - 7. We can get standard form of y = mx + b by simply subtracting 4 from both sides and we get y = 7x - 11. But the form your teacher wants has all the terms on one side of the equation, so by subtracting y from both sides the equation becomes 0 = 7x - y - 11.

Lastly, the point where the line crosses the x-axis is the x-intercept, and all points on the x-axis have a y value of 0. So plug 0 in for y in the equation and solve for x. That gives 0 = 7x - 11. Add 11 to bpoth sides and then divide by 7 to get the x value x = 11/7. The coordinates of the x-intercept then are (11/7,0).

Hope this helps.

Answer 2

difference in y coordinates is 2-1=1

difference in x coordinates is 5-(-2)=5+2=7

square the differences and add them to get 1 + 49 = 50

the square root of 50 is the distance from A to B

the line from A to B has slope = 1/7 = the difference in y values divided by the difference in x values

Then use a point slope equation for the line: y-y1 = m(x-x1) where (x1,y1) is either A or B

y-2 = (1/7)(x-5)

multiply by 7 to get

7y-14 = x -5

add 5 and subtract x from both sides to get

-x+7y-9 = 0

C is on the x-axis where y=0

plug y=0 into the equation

-x + 7(0) -9 = 0

-x -9 = 0

x=-9

y=0

C is (-9,0)

It does help to plot the points, graph it, and draw the line. You can see better what's going on.