Question

what values of v and w make \\( \triangle ijk \cong \triangle yzx \\)?
(there is a right triangle ijk with right angle at j, side ij is w, side ik is v+22; and a right triangle yzx with right angle at z, side yz is 3w-34, side xz is 3v-14. there are input boxes for v = \\( \square \\) and w = \\( \square \\))

Explanation:

Step1: Set up equations for congruent triangles

Since \( \triangle IJK \cong \triangle YZX \), their corresponding sides are equal. So, \( IJ = YZ \) and \( IK = XZ \) (or \( JK = XY \), but let's use the given expressions). From the triangles, \( IJ = w \), \( YZ = 3w - 34 \), \( IK = v + 22 \), and \( XZ = 3v - 14 \). Also, both are right triangles (right angles at \( J \) and \( Z \)). So we have two equations:

1. \( w = 3w - 34 \) (from \( IJ = YZ \))
2. \( v + 22 = 3v - 14 \) (from \( IK = XZ \))

Step2: Solve for \( w \)

From \( w = 3w - 34 \), subtract \( w \) from both sides: \( 0 = 2w - 34 \). Then add 34 to both sides: \( 2w = 34 \). Divide by 2: \( w = 17 \).

Step3: Solve for \( v \)

From \( v + 22 = 3v - 14 \), subtract \( v \) from both sides: \( 22 = 2v - 14 \). Add 14 to both sides: \( 36 = 2v \). Divide by 2: \( v = 18 \).

Answer:

\( v = 18 \), \( w = 17 \)