Question

the black triangle in the first quadrant is the original figure. label each other triangle with its transformation. x - axis reflection y - axis reflection translation of 6 to the left translation of 6 to the left and down 4 translation of 3 to the left and down 6

Explanation:

To solve this, we analyze each triangle's transformation:

Green to Black Triangle:

• Observe the horizontal shift. The green triangle moves right (or the black is a translation). Count the horizontal units: from green’s vertex to black’s, it’s a translation of 3 to the right? Wait, no—wait, original (black in first quadrant). Wait, the original is black in first quadrant. Let's re-express:

1. Original (Black) in 1st Quadrant:

• Let’s track vertices. For the green triangle (left of y-axis), compare to black (right of y-axis). The green to black: horizontal shift (translation) of 3 units right? Wait, no—wait the options include “translation of 3 to the left and down 6” etc. Wait, maybe better to list each:

• Green Triangle (left of y-axis): Compare to black (right). If original is black, then green is a translation? Wait, the problem says “the black triangle in the first quadrant is the original figure. Label each other triangle with its transformation.”

So:

• Green Triangle: Relative to original (black), check shift. If original is black (first quadrant), green is a translation of 6 to the left (since from black’s x-coordinate, moving left 6 units? Wait the options have “translation of 6 to the left”. Let's check coordinates. Suppose original black has a vertex at (3, 3) (approx), green at (-3, 3): that’s 6 left. So green is “translation of 6 to the left”.

• Purple Triangle: Reflect over x-axis? Wait, purple is below x-axis, original above. So x-axis reflection? Wait, original is above x-axis, purple is below, same x, opposite y: x-axis reflection.

• Blue Triangle: Maybe translation or reflection. Wait the red triangle: “translation of 3 to the left and down 6”? Wait, let's match options:

Options:

• X - axis reflection
• Y - axis reflection
• translation of 6 to the left
• translation of 6 to the left and down 4
• translation of 3 to the left and down 6

Step - by - Step Labeling:

1. Green Triangle (left of y - axis):

• Compare to original (black, first quadrant). Horizontal shift: original x to green x is 6 units left (e.g., original x = 3, green x = - 3: 3 - (-3)=6 left). So transformation: translation of 6 to the left.

1. Purple Triangle (below x - axis, left of y - axis):

• Original (black) is above x - axis, purple is below, same horizontal position (x - coordinate), opposite y - coordinate. So reflection over x - axis: X - axis reflection.

1. Blue Triangle (below x - axis, left of y - axis, slanting):

• Wait, blue is a translation? Or reflection? Wait, purple is x - axis reflection, blue: maybe translation of 6 to the left and down 4? Wait, original (black) to blue: left 6, down 4? Let's check y - axis: original y is 3, blue y is - 1: 3 - (-1)=4 down. So translation of 6 to the left and down 4.

1. Red Triangle (below x - axis, right of y - axis):

• Original (black) to red: left 3, down 6? Original y = 3, red y = - 3: 3 - (-3)=6 down; original x = 3, red x = 0: 3 - 0 = 3 left. So translation of 3 to the left and down 6.

Final Labeling (Example for Each Triangle):

• Green: translation of 6 to the left
• Purple: X - axis reflection
• Blue: translation of 6 to the left and down 4
• Red: translation of 3 to the left and down 6

(Note: Exact labeling depends on precise coordinate tracking, but using the given options, this matches the transformations.)

If you need to label one, e.g., the green triangle:

Step1: Identify original (black in 1st quadrant).

Step2: Compare green’s position: horizontal shift left by 6 units (no vertical shift).

Answer:

translation of 6 to the left