How do you determine the ratio of similar triangles?

Answer 1

It depends on what information you have.

If you know only that two of the angles of one triangle are the same as the corresponding angles of the other (and since the third angles are 180 minus these two, they are also the same), you can determine NOTHING about the ratio.

You will need at least one side in each triangle.

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If you have two corresponding sides, then the ratio of the triangles is the ratio of the sides.

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If you have two non-corresponding sides with the opposite angle in each case, you can use the sine rule to determine the ratio as follows:

Triangle ABC with sides a, b and c where a is opposite A, b opposite B and c opposite C.
Triangle PQR, similar to ABC with sides p, q and r with similar opposition.

Suppose you know a, A, q and Q (not p and P since that would be the previous scenario).

Ratio of sides of triangles = b/q = b/sinB*sinB/q (multiply and divide by sinB)
= b/sinB*sinQ/q (B and Q are corresponding angles of the two triangles so they and hence their sines are the same)

= a/sinA * sinQ/q (a/sinA = b/sinB by the sine rule)

All these elements are known and so b/q is determined.

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If a, A, q and R are known then effectively Q is known and we are back to the previous case.

Why is Q known? In triangle PQR, P = A is known. R is known and P+Q+R = 180

Not sure if you can determine the ratio in other cases.