Question

chapter 2 test a
triangles abc and qrs are congruent.

1. which angle of abc corresponds to ∠r?
2. which angle of qrs corresponds to ∠a?
3. which side of abc corresponds to side sq?

tell whether the shaded figure is a translation, reflection, rotation,
or dilation of the non - shaded figure.
4.
5.
6.
7.

1. the vertices of a rectangle are ( a(2,0), b(5,0), c(5, - 2) ), and ( d(2, - 2) ). reflect the rectangle in the ( y ) - axis.

what are the coordinates
of the image?

1. the vertices of a triangle are ( x(-3,3), y(-1,3) ), and ( z(-3,0) ).

dilate the triangle with respect to the origin using a scale factor
of 2.
what are the coordinates of the image?

1. rotate the triangle ( 180^{circ} ) about the origin. find the coordinates of

the image.
answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
see left.

Explanation:

Step1: Determine congruent triangle angles

For congruent triangles \(ABC\) and \(QRS\), the correspondence is based on the order of congruence.

1. In congruent triangles \(ABC\cong QRS\), \(\angle C\) corresponds to \(\angle R\) (by the order of congruence of the triangles).
2. \(\angle Q\) corresponds to \(\angle A\) (by the order of congruence of the triangles).
3. For the side correspondence, side \(AB\) corresponds to side \(SQ\) (by the order of congruence of the triangles).

Step2: Identify transformation types

1. For the first pair of figures (the “W” - like and “E” - like shapes), it is a rotation. A rotation is a transformation where a figure is turned around a fixed point.
2. For the two stars, it is a translation. A translation is a transformation that slides a figure without rotating or resizing it.
3. For the two octagons (one smaller and one larger), it is a dilation. A dilation is a transformation that resizes a figure.
4. For the two lightning - bolt - like shapes, it is a reflection. A reflection is a transformation that flips a figure over a line.

Step3: Reflect rectangle over \(y\) - axis

The rule for reflecting a point \((x,y)\) over the \(y\) - axis is \((x,y)\to(-x,y)\).

• For \(A(2,0)\), \(A'\) is \((- 2,0)\)
• For \(B(5,0)\), \(B'\) is \((-5,0)\)
• For \(C(5,-2)\), \(C'\) is \((-5,-2)\)
• For \(D(2,-2)\), \(D'\) is \((-2,-2)\)

Step4: Dilate triangle with scale factor \(k = 2\)

The rule for dilating a point \((x,y)\) with respect to the origin and scale factor \(k\) is \((x,y)\to(kx,ky)\).

• For \(X(-3,3)\), \(X'\) is \((-6,6)\)
• For \(Y(-1,3)\), \(Y'\) is \((-2,6)\)
• For \(Z(-3,0)\), \(Z'\) is \((-6,0)\)

Step5: Rotate triangle \(180^{\circ}\) about the origin

The rule for rotating a point \((x,y)\) \(180^{\circ}\) about the origin is \((x,y)\to(-x,-y)\).
Assume the coordinates of \(A\), \(B\), \(C\) (from the triangle in problem 10, let \(A(1,1)\), \(B(0,4)\), \(C(3,1)\) - assuming coordinates from the graph).

• \(A(1,1)\to A'(-1,-1)\)
• \(B(0,4)\to B'(0,-4)\)
• \(C(3,1)\to C'(-3,-1)\)

Answer:

1. \(\angle C\)
2. \(\angle Q\)
3. \(AB\)
4. Rotation
5. Translation
6. Dilation
7. Reflection
8. \(A'(-2,0)\), \(B'(-5,0)\), \(C'(-5,-2)\), \(D'(-2,-2)\)
9. \(X'(-6,6)\), \(Y'(-2,6)\), \(Z'(-6,0)\)
10. (Assuming \(A(1,1)\), \(B(0,4)\), \(C(3,1)\)) \(A'(-1,-1)\), \(B'(0,-4)\), \(C'(-3,-1)\)