Question

9 multiple choice 1 point what can we conclude about the triangles shown on the line below? the triangles are congruent and therefore have the same slope the triangles are similar and therefore have the same slope the triangles are similar but do not have the same slope the triangles are unrelated to each other

Explanation:

Step1: Calculate the slopes

For the blue triangle \(AB - BC\), slope \(m_1=\frac{3}{4}\).
For the green triangle \(FG - GH\), slope \(m_2=\frac{2}{ \frac{4}{3}\times2}\) (since the base of green triangle is \(\frac{4}{3}\times2\) when compared to blue triangle's base 4 and height 2 compared to blue triangle's height 3, but more simply, using the ratio of vertical change to horizontal change. If we assume the horizontal change for green triangle is \(x\) and vertical is \(y\), from the figure we can see that if we consider the similar - triangle property. For the large red - outlined triangle \(CD - DE\), slope \(m_3=\frac{6}{8}=\frac{3}{4}\).
Since the triangles are right - angled and the ratios of their corresponding sides are equal (for example, if we assume one triangle has base \(b_1\) and height \(h_1\), another has base \(b_2 = kb_1\) and height \(h_2=kh_1\) where \(k\) is a non - zero constant). The slope of a line segment in a right - angled triangle (which represents the slope of the hypotenuse) is given by \(m=\frac{h}{b}\). If \(\frac{h_1}{b_1}=\frac{h_2}{b_2}\) (because of similar triangles, \(\frac{h_2}{h_1}=\frac{b_2}{b_1}\)), then the slopes are equal.

Step2: Check for similarity

Two right - angled triangles are similar if the ratios of their corresponding sides are equal. Let's assume one triangle has base \(b_1\) and height \(h_1\), another has base \(b_2\) and height \(h_2\). If \(\frac{b_2}{b_1}=\frac{h_2}{h_1}\) (for example, if \(b_1 = 4,h_1=3,b_2 = 8,h_2 = 6\), \(\frac{8}{4}=\frac{6}{3}=2\); if \(b_1 = 4,h_1 = 3,b_2=\frac{4}{3}\times2,h_2 = 2\), \(\frac{\frac{4}{3}\times2}{4}=\frac{2}{3}\) and \(\frac{2}{3}\) (since \(\frac{2}{3}=\frac{2}{3}\)). The triangles are similar by the Side - Angle - Side (SAS) similarity criterion (the right angle is the included angle and the ratios of the sides forming the right angle are equal).

Answer:

The triangles are similar and therefore have the same slope.