In a triangle ABC, D and E are points on BC such that AD =AE and angle BAD = angle CAE, AB = (2p + 3) BD = 2p AC = (3q - 1) and CE = q then find the value of (p + q)
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We have a triangle ABC with points D and E on side BC. We're given that AD equals AE, and angle BAD equals angle CAE. The side lengths are expressed in terms of variables p and q: AB equals 2p plus 3, BD equals 2p, AC equals 3q minus 1, and CE equals q. We need to find the value of p plus q.
Looking at the given conditions, we notice a clear symmetry pattern. We have AD equals AE and angle BAD equals angle CAE. This symmetry suggests that triangle ABD is congruent to triangle ACE. The conditions are symmetric with respect to swapping B with C, D with E, AB with AC, and BD with CE.
If triangle ABD is congruent to triangle ACE, then their corresponding sides must be equal. This gives us two key equations. First, AB equals AC, which means 2p plus 3 equals 3q minus 1. Second, BD equals CE, which means 2p equals q. Now we have a system of two equations with two unknowns.
Now let's solve the system of equations. We have 2p plus 3 equals 3q minus 1, and 2p equals q. We substitute q equals 2p into the first equation: 2p plus 3 equals 3 times 2p minus 1, which simplifies to 2p plus 3 equals 6p minus 1. Rearranging gives us 4 equals 4p, so p equals 1. Therefore q equals 2 times 1, which is 2. Finally, p plus q equals 1 plus 2, which equals 3.
To summarize our solution: We identified the symmetry in the given conditions which suggested triangle congruence. We used the fact that congruent triangles have equal corresponding sides to establish our equations. Solving the system gave us p equals 1 and q equals 2. Therefore, the value of p plus q is 3.