Question

the graph shows triangles jkl and opq. is jkl similar to opq? justify your answer. yes, because a dilation by a scale factor of 2 centered at point j maps jkl onto opq. yes, because a dilation by a scale factor of 2 centered at point l maps jkl onto opq. no, because ∠k and ∠p do not have the same measure.

Explanation:

Step1: Recall similarity by dilation

Two triangles are similar if one is a dilation of the other. A dilation centered at a point multiplies the distances of all points from the center - point by the scale - factor.

Step2: Check dilation from point J

Let's assume a dilation centered at point J. If we consider the coordinates of the vertices of \(\triangle JKL\) and \(\triangle OPQ\) and check the distances from point J. Let \(J=( - 3,1)\), \(K=(0, - 3)\), \(L=( - 3,-2)\), \(O=( - 3,1)\), \(P=(3,-7)\), \(Q=( - 3,-6)\).
The distance between \(J\) and \(K\) in the \(y\) - direction is \(1-( - 3)=4\), and the distance between \(O\) and \(P\) in the \(y\) - direction is \(1-( - 7)=8\). The distance between \(J\) and \(L\) in the \(y\) - direction is \(1-( - 2)=3\), and the distance between \(O\) and \(Q\) in the \(y\) - direction is \(1-( - 6)=7\).
If we consider a dilation centered at \(J\), the \(x\) - coordinate of \(J\) and \(O\) is the same (\(x = - 3\)). For point \(K\) with \(x = 0\) and point \(P\) with \(x = 3\), the horizontal distance from \(J\) to \(K\) is \(|0-( - 3)| = 3\), and the horizontal distance from \(O\) to \(P\) is \(|3-( - 3)|=6\). A dilation by a scale factor of 2 centered at \(J\) maps \(\triangle JKL\) onto \(\triangle OPQ\).

Step3: Recall similarity criteria

If a dilation maps one triangle onto another, the two triangles are similar because corresponding angles are congruent (since dilation is a similarity transformation) and corresponding sides are in proportion.

Answer:

Yes, because a dilation by a scale factor of 2 centered at point \(J\) maps \(JKL\) onto \(OPQ\).