Question

the graph shows triangles opq and vwx. is opq congruent to vwx? justify your answer. yes, because a translation right 13 units and up 1 unit maps opq onto vwx. yes, because a reflection across the y - axis maps opq onto vwx. no, because \\( \overline { op } \\) and \\( \overline { vw } \\) do not have the same length. no, because \\( \overline { pq } \\) and \\( \overline { wx } \\) do not have the same length.

Explanation:

Step1: Calculate the length of \( \overline{PQ} \)

Use the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For points \( P(-6,-3) \) and \( Q(-7,5) \), \( x_1=-6,y_1 = - 3,x_2=-7,y_2 = 5 \).

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Step2: Calculate the length of \( \overline{WX} \)

For points \( W(7,-2) \) and \( X(6,6) \), \( x_1 = 7,y_1=-2,x_2 = 6,y_2 = 6 \)

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Step3: Calculate the length of \( \overline{OP} \)

For points \( O(-8,-4) \) and \( P(-6,-3) \), \( x_1=-8,y_1=-4,x_2=-6,y_2=-3 \)

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Step4: Calculate the length of \( \overline{VW} \)

For points \( V(5,-3) \) and \( W(7,-2) \), \( x_1 = 5,y_1=-3,x_2 = 7,y_2=-2 \)

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Step5: Check congruence conditions

Two triangles are congruent if their corresponding sides are equal.

• \( OP=\sqrt{5},VW=\sqrt{5} \)
• \( PQ=\sqrt{65},WX=\sqrt{65} \)
• \( OQ=\sqrt{(-7 + 8)^2+(5 + 4)^2}=\sqrt{1+81}=\sqrt{82} \), \( VX=\sqrt{(6 - 5)^2+(6+3)^2}=\sqrt{1 + 81}=\sqrt{82} \)

Since all corresponding sides are equal, triangles \( OPQ \) and \( VWX \) are congruent. A translation right \( 13 \) units (\( - 8+13 = 5,-6+13 = 7,-7+13 = 6 \)) and up \( 1 \) unit (\( -4 + 1=-3,-3+1=-2,5+1 = 6 \)) maps \( OPQ \) onto \( VWX \)

Answer:

Yes, because a translation right 13 units and up 1 unit maps \( OPQ \) onto \( VWX \)