Question

bonus: determine the coordinates of point k and give the scale factor.
n(-24, 0) l(-15, 0) o m(0, 16)
k= ( ) sf

Explanation:

Step1: Identify Similar Triangles

The triangles are similar (by AA similarity, as they share the same angle at the origin and both have right angles with the x - axis). For similar triangles, the ratio of corresponding sides is equal (scale factor). The x - coordinates of the vertices on the x - axis are \(N(-24,0)\) and \(L(-15,0)\), and the y - coordinate of \(M\) is \(16\). Let the y - coordinate of \(K\) be \(y\). The ratio of the lengths of \(OL\) to \(ON\) (where \(O\) is the origin \((0,0)\)) is \(\frac{|-15|}{|-24|}=\frac{15}{24}=\frac{5}{8}\).

Step2: Find Coordinates of K

Since the triangles are similar, the ratio of the y - coordinates (height) will be the same as the ratio of the x - coordinates (base). The y - coordinate of \(M\) is \(16\), so the y - coordinate of \(K\) is \(16\times\frac{5}{8} = 10\). The x - coordinate of \(K\) is \(0\) (since it's on the y - axis? Wait, no, looking at the graph, \(K\) is on the line from \(L\) to the y - axis? Wait, maybe the x - coordinate of \(K\) is \(0\)? Wait, no, the points \(N(-24,0)\), \(L(-15,0)\), and \(M(0,16)\) (assuming \(M\) is \((0,16)\) as per the y - axis). Then the line from \(N(-24,0)\) to \(M(0,16)\) and the line from \(L(-15,0)\) to \(K\). The slope of \(NM\) is \(\frac{16 - 0}{0-(-24)}=\frac{16}{24}=\frac{2}{3}\). The equation of line \(NM\) is \(y=\frac{2}{3}(x + 24)\). For point \(L(-15,0)\), we want to find the point \(K\) on the line parallel? No, similar triangles. The ratio of \(OL\) to \(ON\) is \(\frac{15}{24}=\frac{5}{8}\). So the coordinates of \(K\) can be found by scaling the coordinates of \(M\) by the scale factor \(\frac{5}{8}\). Since \(M\) is \((0,16)\), wait, no, \(K\) is on the y - axis? Wait, maybe the x - coordinate of \(K\) is \(0\), and y - coordinate is \(16\times\frac{15}{24}=10\). So \(K=(0,10)\).

Step3: Find Scale Factor

The scale factor (sf) is the ratio of the corresponding sides. The length of \(OL = 15\) (distance from \(O\) to \(L\)) and \(ON = 24\) (distance from \(O\) to \(N\)). So \(sf=\frac{15}{24}=\frac{5}{8}\) (or \(\frac{10}{16}=\frac{5}{8}\) using the y - coordinates).

Answer:

Coordinates of \(K\) are \((0,10)\) and scale factor \(sf=\frac{5}{8}\)