Use the image to answer the question.

Question

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
?

Answer 1

To find the segment on triangle DEF that is congruent to the segment BC of triangle ABC, we start by analyzing the segments in each triangle.

1. Identify the coordinates of points B and C in triangle ABC:

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

2. Calculate the length of segment BC: Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \): \[ BC = \sqrt{(6 - 3)^2 + (2 - 6)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. Determine the coordinates of points D and E in triangle DEF after reflecting ABC across the y-axis:

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

4. Identify the segment DE in triangle DEF: Using the distance formula again for segment DE: \[ DE = \sqrt{(-3 - (-3))^2 + (6 - 2)^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{16} = 4 \]

5. Identify the other segments in triangle DEF: Next, we will consider other segments:

   • Segment EF: Points E \( (-3, 6) \) and F \( (-6, 2) \): \[ EF = \sqrt{(-6 - (-3))^2 + (2 - 6)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

   • Segment FD: Points F \( (-6, 2) \) and D \( (-3, 2) \): \[ FD = \sqrt{(-3 - (-6))^2 + (2 - 2)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3 \]

6. Identify the congruent segment: The segment EF in triangle DEF is congruent to segment BC in triangle ABC, both having a length of 5.

Thus, the segment in triangle DEF that is congruent to segment BC of triangle ABC is segment EF.