Question

undefined term in geometry. use the number line for items 6 - 9. 6. what is ab? 7. what is the coordinate of the midpoint of $overline{ab}$? explain how you found your answer. 8. point m is the midpoint of $overline{ab}$. what is the coordinate of the midpoint of $overline{am}$? 9. point c is between points a and b. the distance between points b and c is $\frac{1}{4}$ of ab. what is the coordinate of point c? 10. $overline{fg}$ lies on a number line. the coordinate of point f is 8. given that $fg = 16$, what are the two possible coordinates for point g?

Explanation:

Step1: Find the distance \(AB\)

The coordinate of \(A=- 12\) and the coordinate of \(B = 8\). The distance between two points \(x_1\) and \(x_2\) on a number - line is \(d=\vert x_2 - x_1\vert\). So \(AB=\vert8-(-12)\vert=\vert8 + 12\vert=20\).

Step2: Find the mid - point of \(AB\)

The formula for the mid - point of two points \(x_1\) and \(x_2\) on a number - line is \(M=\frac{x_1 + x_2}{2}\). Here \(x_1=-12\) and \(x_2 = 8\), so \(M=\frac{-12 + 8}{2}=\frac{-4}{2}=-2\).

Step3: Find the mid - point of \(AM\)

Since \(M\) is the mid - point of \(AB\) and \(M=-2\), \(A=-12\). Using the mid - point formula \(M'=\frac{-12+( - 2)}{2}=\frac{-12-2}{2}=\frac{-14}{2}=-7\).

Step4: Find the coordinate of point \(C\)

We know \(AB = 20\), and \(BC=\frac{1}{4}AB=\frac{1}{4}\times20 = 5\). Since \(B = 8\), and \(C\) is between \(A\) and \(B\), then the coordinate of \(C\) is \(8-5 = 3\).

Step5: Find the possible coordinates of \(G\)

If \(FG = 16\) and \(F = 8\), then if \(G\) is to the right of \(F\), \(G=8 + 16=24\); if \(G\) is to the left of \(F\), \(G=8-16=-8\).

Answer:

1. \(AB = 20\)
2. The coordinate of the mid - point of \(AB\) is \(-2\). We used the mid - point formula \(M=\frac{x_1 + x_2}{2}\) with \(x_1=-12\) and \(x_2 = 8\).
3. The coordinate of the mid - point of \(AM\) is \(-7\).
4. The coordinate of point \(C\) is \(3\).
5. The two possible coordinates for point \(G\) are \(24\) and \(-8\).