The coordinates of the vertices of CDE are C(1, 4), D(3, 6), and E(7, 4). If the triangle is reflected over the line y = 3, what are the coordinates of the image of D?
(3, –6)
(3, –3)
(3, 0)
(3, 9)
i have no clue
C. 3, 0
Well, well, well! Looks like we've got ourselves a reflection problem. Let's dive right in, shall we?
Since we're reflecting the triangle over the line y = 3, we just need to find the image of D. Now, when you reflect a point, the x-coordinate stays the same, but the y-coordinate gets flipped. It's like giving someone a high-five and turning it into a low-five!
So, the x-coordinate of point D is 3, which remains unchanged. And the y-coordinate of D is 6. But since we're reflecting over the line y = 3, we need to flip that y-coordinate to the other side of the line.
Drumroll, please! 🥁
The image of D is at coordinates (3, -6)! It's like D got surprised and ended up upside down. Bet that gave D quite the reflection shock!
So there you go, my friend. The correct answer is (3, -6). Keep up the good work, and remember, always stay positive (unless you're being reflected)!
To find the image of a point after reflection over a line, we can use the concept of symmetry. In this case, the line of reflection is y = 3.
First, we need to find the distance between the point D(3, 6) and the line y = 3. The distance between a point (x1, y1) and a line Ax + By + C = 0 can be calculated using the formula:
Distance = |Ax1 + By1 + C| / √(A^2 + B^2)
In this case, the equation of the line is y = 3, which can be rewritten as y - 3 = 0. Comparing this to the general equation Ax + By + C = 0, we have A = 0, B = 1, and C = -3.
Using the formula, the distance between D(3, 6) and y = 3 is:
|0*3 + 1*6 - 3| / √(0^2 + 1^2) = |6 - 3| / √(0 + 1) = 3 / 1 = 3.
Now, we can determine the image of D after reflection over the line y = 3. Since the distance between D and the line is 3, the reflected image of D will also be located 3 units on the other side of the line.
Since D lies above the line y = 3, the image of D will be below the line y = 3. Thus, the y-coordinate of the image will be 3 - 3 = 0.
Therefore, the coordinates of the image of D are (3, 0).
So, the correct answer is (3, 0).
To find the image of point D after reflecting the triangle over the line y = 3, we can use the concept of reflection.
First, let's calculate the distance between point D and the line y = 3. The line y = 3 is parallel to the x-axis and is 3 units away. Thus, the distance between point D and the line y = 3 is |6 - 3| = 3 units.
Next, we need to determine if the image of D will be above or below the line y = 3. Since the distance between D and the line y = 3 is 3 units, and the y-coordinate of D is 6, which is 3 units above the line y = 3, the image of D will be 3 units below the line y = 3.
Now, to find the coordinates of the image of D, we need to maintain the same x-coordinate and reflect the y-coordinate with respect to the line y = 3. Since D(3, 6) is above the line y = 3, the image of D will be 3 units below the line y = 3. Therefore, the y-coordinate of the image of D will be 3 - 3 = 0.
Thus, the coordinates of the image of D are (3, 0).
Therefore, the correct answer is (3, 0).