Imagine two scalene triangles. One has the Point M on its top corner, the bottom left corner Point K, and the bottom right corner Point L. On the bottom line is "20", the left line is "10", and the right line is "15". The other triangle, has Point R on its top corner, Point P on its left corner, and Point Q on its bottom right corner. The numbers would be "12" on the top left line, "18" on the bottom left line, and "24" on the right line.

Are these two triangles similar? If so, why? Make sure to explain your answer and what similarity theorem you can use (AA, SAS, SSS). If similar, also create a similarity statement.

1 answer

To determine whether the two triangles are similar, we need to analyze the side lengths of both triangles and check the ratios of the corresponding sides.

Triangle 1 (M-K-L):

  • Side MK (left) = 10
  • Side KL (bottom) = 20
  • Side ML (right) = 15

Triangle 2 (R-P-Q):

  • Side RP (left) = 12
  • Side PQ (bottom) = 18
  • Side RQ (right) = 24

Next, we can calculate the ratios of the corresponding sides:

For Triangle 1:

  • \( \text{MK} : \text{KL} : \text{ML} = 10 : 20 : 15 \)

For Triangle 2:

  • \( \text{RP} : \text{PQ} : \text{RQ} = 12 : 18 : 24 \)

Now, let's simplify both sets of ratios:

Triangle 1 Ratios:

  • MK:KL = \( \frac{10}{20} = \frac{1}{2} \)
  • KL:ML = \( \frac{20}{15} = \frac{4}{3} \)
  • ML:MK = \( \frac{15}{10} = \frac{3}{2} \)

Triangle 2 Ratios:

  • RP:PQ = \( \frac{12}{18} = \frac{2}{3} \)
  • PQ:RQ = \( \frac{18}{24} = \frac{3}{4} \)
  • RQ:RP = \( \frac{24}{12} = 2 \)

To compare the triangles, we can compute the side length ratios as follows:

  • MK : KL : ML = 10 : 20 : 15 = \( 2 : 4 : 3 \) (after reducing)
  • RP : PQ : RQ = 12 : 18 : 24 = \( 2 : 3 : 4 \) (after reducing)

Now, we can see that the ratios of the sides are not the same.

Conclusion:

Since the ratios are not equivalent, the two triangles are not similar. The side ratios do not maintain a consistent proportionality factor, which is required for similarity by either AA (Angle-Angle) or SSS (Side-Side-Side).

Final Statement:

Triangles M-K-L and R-P-Q are not similar, as their corresponding side ratios (10:20:15 and 12:18:24) do not match proportionally.

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