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
?

(1 point)
segment

Answer 1

To determine which segment on triangle DEF is congruent to segment BC in triangle ABC, we first need to find the coordinates of the points forming segment BC.

In triangle ABC:

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

To find the length of segment BC, we can use the distance formula:

\[ \text{Length of } BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of points B and C:

\[ \text{Length of } BC = \sqrt{(6 - 3)^2 + (2 - 6)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Now, let's examine triangle DEF:

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

To find the segment in triangle DEF that is congruent to segment BC, we examine the segments DE, EF, and FD.

1. Segment DE (from D to E):

   • Length DE = \(\sqrt{((-3) - (-3))^2 + (6 - 2)^2} = \sqrt{0^2 + (4)^2} = \sqrt{16} = 4\)

2. Segment EF (from E to F):

   • Length EF = \(\sqrt{((-6) - (-3))^2 + (2 - 6)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

3. Segment FD (from F to D):

   • Length FD = \(\sqrt{((-3) - (-6))^2 + (2 - 2)^2} = \sqrt{(3)^2 + 0^2} = \sqrt{9} = 3\)

Comparing the lengths:

• Length DE = 4
• Length EF = 5
• Length FD = 3

The segment in triangle DEF that is congruent to segment BC (which has length 5) is segment EF.

Therefore, the final answer is:

segment EF