Question

what values of s and t make \\( \triangle b c d \cong \triangle h f g \\) ?

Explanation:

Step1: Set up equations based on congruent sides

Since \(\triangle BCD\cong\triangle HFG\), we have \(BC = HG\) and \(CD=GF\).
So, \(-s + t+90=29t - 2s-83\) and \(-s + t + 40=5s-13t\).

Step2: Simplify the first equation

\(-s + t+90=29t - 2s-83\)
\(-s+2s+t - 29t=-83 - 90\)
\(s-28t=-173\) (Equation 1)

Step3: Simplify the second equation

\(-s + t + 40=5s-13t\)
\(-s-5s+t + 13t=-40\)
\(-6s+14t=-40\)
Divide by \(2\): \(-3s + 7t=-20\) (Equation 2)

Step4: Solve the system of equations

From Equation 1: \(s=28t - 173\)
Substitute \(s = 28t-173\) into Equation 2:
\(-3(28t - 173)+7t=-20\)
\(-84t + 519+7t=-20\)
\(-77t=-20 - 519\)
\(-77t=-539\)
\(t = 7\)

Step5: Find the value of \(s\)

Substitute \(t = 7\) into \(s=28t - 173\)
\(s=28\times7-173\)
\(s=196 - 173\)
\(s = 23\)

Answer:

\(s = 23\), \(t = 7\)