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
Remaining Attempts : 3

Answer 1

To find which segment on triangle DEF is congruent to the segment BC in triangle ABC, we first need to identify the coordinates of points B and C.

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

Now, let's calculate the length of segment BC.

The length of a segment formed by two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of 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, we find the corresponding points in triangle DEF after reflecting triangle ABC across the y-axis.

The reflection of a point \((x, y)\) across the y-axis is given by \((-x, y)\).

So, the reflected points are:

• Point D (reflection of A (3, 2)): \((-3, 2)\)
• Point E (reflection of B (3, 6)): \((-3, 6)\)
• Point F (reflection of C (6, 2)): \((-6, 2)\)

Next, we find segment DE which connects points D and E. The coordinates of D and E are:

• D: (-3, 2)
• E: (-3, 6)

Now, we calculate the length of segment DE: \[ \text{Length of } DE = \sqrt{(-3 - (-3))^2 + (6 - 2)^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{16} = 4 \]

Next, we find segment EF which connects points E and F. The coordinates of E and F are:

• E: (-3, 6)
• F: (-6, 2)

Now, we calculate the length of segment EF: \[ \text{Length of } EF = \sqrt{(-6 - (-3))^2 + (2 - 6)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Finally, we find segment FD which connects points F and D. The coordinates of F and D are:

• F: (-6, 2)
• D: (-3, 2)

Now, we calculate the length of segment FD: \[ \text{Length of } FD = \sqrt{(-3 - (-6))^2 + (2 - 2)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3 \]

To summarize:

• Length of segment BC = 5
• Length of segment DE = 4
• Length of segment EF = 5 (congruent to BC)
• Length of segment FD = 3

Thus, the segment on triangle DEF that is congruent to segment BC is: segment EF.