Question

the endpoints of the directed line segment nw are n(-14,9) and w(10, -6). determine point d such that line segment nw is partitioned in a ratio of 1:2. what construction is the image below? bisecting a line segment copying a line segment copying an angle bisecting an angle

Explanation:

1. First sub - question (finding point D that partitions the line segment)

Step1: Recall the section formula

If a point \(D(x,y)\) divides the line - segment joining \(N(x_1,y_1)\) and \(W(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-14,y_1 = 9,x_2 = 10,y_2=-6,m = 1,n = 2\).

Step2: Calculate the x - coordinate of D

\[

$$\begin{align*} x&=\frac{1\times10+2\times(-14)}{1 + 2}\\ &=\frac{10-28}{3}\\ &=\frac{-18}{3}\\ &=-6 \end{align*}$$

\]

Step3: Calculate the y - coordinate of D

\[

$$\begin{align*} y&=\frac{1\times(-6)+2\times9}{1 + 2}\\ &=\frac{-6 + 18}{3}\\ &=\frac{12}{3}\\ &=4 \end{align*}$$

\]

In the given image, arcs are drawn from points \(Q\) and \(R\) on the sides of \(\angle PQR\) and then from the intersection of these arcs, another arc is drawn to create a new angle. This is the process of copying an angle.

Answer:

The coordinates of point \(D\) are \((-6,4)\)

2. Second sub - question (identifying the construction)