Question

3 fill in the blank 1 point what is the coordinate of point p that partitions $overline{hm}$ into the ratio 3:1?

Explanation:

Step1: Recall the section - formula for one - dimensional case

Let the coordinate of \(H\) be \(x_1\) and the coordinate of \(M\) be \(x_2\). The formula for the point \(P\) that divides the line - segment from \(x_1\) to \(x_2\) in the ratio \(m:n\) is \(P=\frac{mx_2+nx_1}{m + n}\). Here, assume \(H\) has coordinate \(x_1\) and \(M\) has coordinate \(x_2\), and the ratio \(m:n = 3:1\).

Step2: Let \(H\) be at \(x_1\) and \(M\) be at \(x_2\)

If we assume the number line is a one - dimensional coordinate system, and let the coordinate of \(H\) be \(a\) and the coordinate of \(M\) be \(b\). Then the coordinate of \(P\) is \(P=\frac{3b + 1a}{3+1}=\frac{3b + a}{4}\). Since we are not given the actual values of \(a\) and \(b\), if we assume \(H\) is at \(x = h\) and \(M\) is at \(x = m\), the coordinate of \(P\) is \(P=\frac{3m+h}{4}\). If we assume \(H\) is at \(2\) and \(M\) is at \(10\):
\[P=\frac{3\times10 + 2}{4}=\frac{30 + 2}{4}=\frac{32}{4}=8\]
In general, if \(H\) is at \(h\) and \(M\) is at \(m\), the formula for \(P\) is \(P=\frac{3m + h}{4}\).

Answer:

\(\frac{3\times(\text{coordinate of }M)+(\text{coordinate of }H)}{4}\)