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An illustration shows a coordinate plane. The x-axis and the y-axis range from negative 6 to 6 in one unit increments. Six points are plotted on the plane. Two right triangles are graphed on the plane. Three points are plotted in the first quadrant. Point A is plotted at the coordinates left parenthesis 3 comma 2 right parenthesis, Point B is plotted at left parenthesis 3 comma 6 right parenthesis, and point C is plotted at left parenthesis 6 comma 2 right parenthesis. Points A and B, points B and C, and points C and A are connected by a dotted line segment to form triangle A B C. Three points are plotted in the second quadrant. Point D is plotted at the coordinates left parenthesis negative 3 comma 2 right parenthesis, Point E is plotted at left parenthesis negative 3 comma 6 right parenthesis, and point F is plotted at left parenthesis negative 6 comma 2 right parenthesis. Points D and E, points E and F, and points F and D are connected by line segments to form triangle D E F.

Triangle ABC is reflected across the y-axis to produce triangle DEF. Which segment on DEF is congruent to the segment BC?

segment __

1 answer

To determine which segment on triangle DEF is congruent to segment BC from triangle ABC, we need to first identify the coordinates of the points in both triangles.

For triangle ABC:

  • Point A is at \( (3, 2) \)
  • Point B is at \( (3, 6) \)
  • Point C is at \( (6, 2) \)

Now, let's find the length of segment BC:

  • The coordinates of points B and C are \( (3, 6) \) and \( (6, 2) \), respectively.
  • The length of segment BC can be calculated using the distance formula: \[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of B and C: \[ BC = \sqrt{(6 - 3)^2 + (2 - 6)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Now, we look at triangle DEF, which is the reflection of triangle ABC across the y-axis:

  • Point D is at \( (-3, 2) \)
  • Point E is at \( (-3, 6) \)
  • Point F is at \( (-6, 2) \)

Now, we check the segments of triangle DEF:

  1. Segment DE: \[ \text{Length of DE} = \sqrt{(-3 - (-3))^2 + (6 - 2)^2} = \sqrt{0^2 + 4^2} = \sqrt{16} = 4 \]
  2. Segment EF: \[ \text{Length of EF} = \sqrt{(-6 - (-3))^2 + (2 - 6)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
  3. Segment FD: \[ \text{Length of FD} = \sqrt{(-3 - (-6))^2 + (2 - 2)^2} = \sqrt{(3)^2 + 0^2} = \sqrt{9} = 3 \]

After checking the lengths of the segments in triangle DEF, we find segment EF has a length of 5, which is congruent to segment BC.

Therefore, the answer is: segment EF.