Question

1. triangle abc is shown on the coordinate grid below. what is the dilation factor that occurred? 4) quadrilateral abcd and its image, abcd, are shown on the coordinate grid below. which rule best describes the transformation that was used to form quadrilateral abcd? a) (x, y)→(−y, x) b) (x, y)→(x, −y) c) (x, y)→(y, x) d) (x, y)→(−y, −x) 5) which 2 figures do not have a horizontal line of symmetry? a) image of yin - yang symbol b) image of a wheel - like symbol c) image of a knot - like symbol d) image of a triangle

Explanation:

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Step1: Notice corresponding side lengths

Measure side - lengths of the original triangle \(ABC\) and the dilated triangle \(A'B'C'\) (using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) if needed). Let's assume we find two corresponding sides. For example, if a side of the original triangle has length \(l_1\) and the corresponding side of the dilated triangle has length \(l_2\).

Step2: Calculate dilation factor

The dilation factor \(k=\frac{l_2}{l_1}\). Without specific coordinates, we can also count the grid - units of corresponding sides. If the original side spans \(a\) grid - units and the dilated side spans \(b\) grid - units, then \(k = \frac{b}{a}\).

Step1: Select a point

Let's take a point \(A(x,y)\) in the original quadrilateral \(ABCD\) and its corresponding point \(A'(x',y')\) in the image \(A'B'C'D'\).

Step2: Test transformation rules

Check each transformation rule:

• For rule a) \((x,y)\to(-y,x)\), if we substitute the coordinates of \(A\) into this rule, we get new coordinates. Compare these new coordinates with the coordinates of \(A'\).
• For rule b) \((x,y)\to(x, - y)\), substitute the coordinates of \(A\) and check.
• For rule c) \((x,y)\to(y,x)\), substitute the coordinates of \(A\) and check.
• For rule d) \((x,y)\to(-y,-x)\), substitute the coordinates of \(A\) and check.

Step1: Recall horizontal line of symmetry concept

A horizontal line of symmetry divides a figure into two congruent parts such that one part is the mirror - image of the other with respect to a horizontal line.

Step2: Analyze each figure

• For figure a) (yin - yang symbol), it has no horizontal line of symmetry.
• For figure b) (wheel - like figure), it may or may not have a horizontal line of symmetry depending on its design details.
• For figure c) (a symmetric figure), it likely has a horizontal line of symmetry.
• For figure d) (equilateral triangle), it has no horizontal line of symmetry.

Answer:

(No specific answer can be given without side - length measurements or grid - unit counts)

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