Question

lesson 4: practice problems grade 8 mathematics 1. triangle abc is dilated using d as the center of dilation with scale factor 2. the image is triangle abc. clare says the two triangles are congruent, because their angle measures are the same. do you agree? explain how you know. 2. on graph paper, sketch the image of quadrilateral pqrs under the following dilations: - the dilation centered at r with scale factor 2. - the dilation centered at r with scale factor 0.5.

Explanation:

1.

Step1: Recall dilation property

Dilation changes side - lengths.

Step2: Recall congruence criteria

For two triangles to be congruent, all corresponding sides and angles must be equal.

Step3: Analyze given triangles

In dilation of $\triangle ABC$ to $\triangle A'B'C'$ with scale factor 2, side - lengths of $\triangle A'B'C'$ are twice those of $\triangle ABC$. Although angle measures are equal (since dilation preserves angle measures), side - lengths are not equal. So the triangles are not congruent.

Step1: Recall dilation formula

For a point \(P(x,y)\) dilated about a center \(C(a,b)\) with scale factor \(k\), the new point \(P'(x',y')\) is given by \(x'=a + k(x - a)\) and \(y'=b + k(y - b)\).

Step2: Apply to vertices of \(PQRS\)

Let the coordinates of \(P(x_1,y_1)\), \(Q(x_2,y_2)\), \(R(x_3,y_3)\), \(S(x_4,y_4)\). For each vertex, use the formula with \(a = x_3\), \(b = y_3\) and \(k = 2\). For example, for vertex \(P\), \(x_{P'}=x_3+2(x_1 - x_3)=2x_1 - x_3\) and \(y_{P'}=y_3+2(y_1 - y_3)=2y_1 - y_3\). Plot the new vertices \(P'\), \(Q'\), \(R'\) (where \(R'=R\) since it is the center of dilation), \(S'\) and connect them to get the dilated quadrilateral.
Dilation centered at \(R\) with scale factor \(0.5\):

Step1: Recall dilation formula

Use the formula \(x'=a + k(x - a)\) and \(y'=b + k(y - b)\) with \(k = 0.5\) and center \(C = R(x_3,y_3)\).

Step2: Apply to vertices of \(PQRS\)

For each vertex \(P(x_1,y_1)\), \(Q(x_2,y_2)\), \(S(x_4,y_4)\) (again \(R'\) is the same as \(R\)), calculate the new coordinates. For vertex \(P\), \(x_{P'}=x_3+0.5(x_1 - x_3)=0.5x_1 + 0.5x_3\) and \(y_{P'}=y_3+0.5(y_1 - y_3)=0.5y_1+0.5y_3\). Plot the new vertices and connect them to get the dilated quadrilateral.

Answer:

No, I do not agree. Dilation changes side - lengths. Congruent triangles require equal side - lengths and equal angle measures. Here, side - lengths are not equal.

2.

Dilation centered at \(R\) with scale factor 2: