What is the ratio between the sides of the triangles ΔABC and ΔMNA?
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What is the ratio between the sides of the triangles ΔABC and ΔMNA?
From the data in the drawing, it seems that angle M is equal to angle B
Also, angle A is an angle shared by both triangles ABC and AMN
That is, triangles ABC and AMN are similar respectively according to the angle-angle theorem.
According to the letters, the sides that are equal to each other are:
Now we can calculate the ratio between the sides of the given triangles:
Is the similarity ratio between the three triangles equal to one?
Look at the angle markings and position of vertices! In triangles ABC and MNA, angle A is shared, and the angle marks show angle B equals angle M. This means BC corresponds to MN.
The ratio means triangle ABC is twice as large as triangle MNA. Every corresponding side of ABC is 2 times longer than the matching side of MNA.
These triangles are similar by AA (Angle-Angle) similarity! They share angle A, and the angle marks show angle B = angle M. Two equal angles is enough to prove similarity.
No! Once you prove triangles are similar, all corresponding sides have the same ratio. Finding one ratio like tells you the ratio for the entire triangles.
The method stays the same! Always identify corresponding sides, divide the lengths in the correct order, and simplify. The ratio might be , , or any other fraction.
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