Question

1. △hij ≅ △pqr. solve for x. then find the missing sides.

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle HIJ\cong\triangle PQR\), corresponding sides are equal. Assume \(HI = PQ\) (by the order of congruence \(\triangle HIJ\cong\triangle PQR\), \(H\) corresponds to \(P\), \(I\) corresponds to \(Q\), \(J\) corresponds to \(R\)). So \(3x=x + 64\).

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides of \(3x=x + 64\):
\(3x-x=x + 64-x\)
\(2x=64\)
Divide both sides by \(2\): \(x = 32\).

Step3: Find the length of \(HI\)

Substitute \(x = 32\) into \(HI=3x\). Then \(HI=3\times32 = 96\).

Step4: Find the length of \(IJ\)

Substitute \(x = 32\) into \(IJ=x + 32\). Then \(IJ=32+32=64\).

Step5: Find the length of \(QR\)

Substitute \(x = 32\) into \(QR = 2x\). Then \(QR=2\times32=64\).

Step6: Verify the side - length correspondence

\(HI = PQ=96\), \(IJ=QR = 64\), \(HJ=PR\) (by congruence, and we can check using another pair of sides if needed. For example, if we assume \(IJ = QR\), \(x + 32=2x\), \(32=x\) which is consistent with our previous result).

Answer:

\(x = 32\), \(HI=96\), \(IJ = 64\), \(QR=64\)