Question 11 of 25intermediate🔍 AnalyzeShort Answer2 marks

Would these observations be the same if ΔADM\Delta ADM is reflected in the xx-axis (instead of the yy-axis)?

Correct Answer

Yes, these observations would be the same. If ΔADM\Delta ADM is reflected in the xx-axis, the x-coordinates of the vertices will remain unchanged, while the y-coordinates will change their signs.

The distance between any two points is calculated using the formula:

(x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Since the differences in the coordinates are squared, the change in sign of the yy-coordinates will not affect the overall distance calculation.

Therefore, just as with reflection in the yy-axis, reflection in the xx-axis will also result in the same side lengths. The observation that reflection preserves the lengths of the sides of the triangles remains unchanged.

Exercise: Think and Reflect | Q: 2 | (Page 11)
For More Understanding

Explanation

The textbook context establishes that reflection in the yy-axis preserves the lengths of the sides of ΔADM\Delta ADM. When reflecting across the xx-axis instead, the yy-coordinates change sign while xx-coordinates remain the same.

Because the distance formula relies on the square of the differences between coordinates, the sign change does not alter the resulting lengths. Thus, the observation regarding the preservation of side lengths holds true for reflection in the xx-axis as well.

Solution Steps

  1. Step 1: Identify that reflecting in the xx-axis changes the sign of the yy-coordinates while keeping xx-coordinates the same.

  2. Step 2: Recall the distance formula:

    (x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

  3. Step 3: Note that squaring the coordinate differences eliminates the effect of the sign change in the yy-coordinates.

  4. Step 4: Conclude that the side lengths remain the same, so the observation that reflection preserves lengths is unchanged.

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