Question

Which statement explains the relationship between the slope of the line and features of the triangles? Responses The triangles are similar and have the same ratios for the height to the base (32=64) ( 3 2 = 6 4 ) . The slope of the line is 32 3 2 . The triangles are similar and have the same ratios for the height to the base ( 3 2 = 6 4 ) . The slope of the line is 3 2 . The triangles are congruent and have the same ratios for the height to the base (32=64) ( 3 2 = 6 4 ) . The slope of the line is 32 3 2 . The triangles are congruent and have the same ratios for the height to the base ( 3 2 = 6 4 ) . The slope of the line is 3 2 . The triangles are congruent and have the same ratios for the height to the base (23=46) ( 2 3 = 4 6 ) . The slope of the line is 23 2 3 . The triangles are congruent and have the same ratios for the height to the base ( 2 3 = 4 6 ) . The slope of the line is 2 3 . The triangles are similar and have the same ratios for the height to the base (23=46) ( 2 3 = 4 6 ) . The slope of the line is 23 2 3 .

Answer 1

The slope of a line represents a constant ratio of 'rise over run', and this ratio corresponds to the proportional sides of similar right-angled triangles on a graph. The term 'similar' (not 'congruent') correctly describes these triangles because their shapes are the same but their sizes may differ.

Step-by-step explanation:

The relationship between the slope of a line and the features of triangles can be understood in terms of similarity and proportionality. When triangles are similar, they have the same shape, which means corresponding angles are equal and the sides are proportional. The slope of a line, which is calculated as the 'rise over run' or the change in y over the change in x, can represent the ratio of the height to the base in similar right-angled triangles formed between the x-axis and the line on a graph.

In the context provided by the figures, if we are given that the slope of the line is 3, it signifies a vertical rise of 3 units for every 1 unit increase horizontally. If triangles are drawn on this graph, with their bases along the x-axis and their heights along the y-axis, each corresponding side of the triangles will be in the ratio of 3:1, reflecting the slope of the line. This constant ratio aligns with the definition of similar triangles which maintain the same proportions between corresponding sides.

It is important to note that the term 'congruent' refers to figures that are identical in shape and size, while 'similar' refers to figures that have the same shape but may differ in size. Since the slope gives us a proportion that applies to any triangle formed along the line, regardless of its size, it is more accurate to say the triangles are similar rather than congruent. The slope does not tell us about the absolute size of the triangles, just the ratio of their sides. Therefore, the correct statement regarding the slope of the line and features of the triangles would be that the triangles are similar and have the same ratios for their height to the base, with the slope indicating this constant ratio.