Question Video: Solving Problems Related to a Reflected Triangle Mathematics • 11th Grade

In the following figure, △𝐴′𝐵′𝐶′ is the image of △𝐴𝐵𝐶 by reflection in the line 𝐿. (1) Fill in the blanks: The length of 𝐴′𝐶′ = _ cm, and the length of 𝐴′𝐵′ = _ cm. (2) Fill in the blanks: Line segment 𝐴𝐴′ is _ to line segment 𝐵𝐵′, and line segment 𝐶𝐶′ is _ to line 𝐿. (3) Find the measure of ∠𝐴.

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Video Transcript

In the following figure, triangle 𝐴 prime 𝐵 prime 𝐶 prime is the image of triangle 𝐴𝐵𝐶 by reflection in the line 𝐿. (1) Fill in the blanks. The length of 𝐴 prime 𝐶 prime equals blank centimeters, and the length of 𝐴 prime 𝐵 prime equals blank centimeters. (2) Fill in the blanks. Line segment 𝐴𝐴 prime is blank to line segment 𝐵𝐵 prime, and line segment 𝐶𝐶 prime is blank to line 𝐿. (3) Find the measure of angle 𝐴.

Remember, when we reflect a polygon in a mirror line, we create a second congruent polygon. This means that the two triangles in our diagram are congruent. That in turn means that their line segments and angle measures are equal. This fact helps us to answer part (1). Line segment 𝐴𝐶 is congruent to line segment 𝐴 prime 𝐶 prime. They must have the same lengths. Since line segment 𝐴𝐶 is four centimeters, line segment 𝐴 prime 𝐶 prime must also be four centimeters. And we put four in the first blank space.

Next, line segment 𝐴𝐵 must be congruent to line segment 𝐴 prime 𝐵 prime. And so, 𝐴 prime 𝐵 prime must be six centimeters in length. And six goes in our second blank space.

Let’s now consider question (2). First, we add line segments 𝐴𝐴 prime and 𝐵𝐵 prime to the diagram. We know that these line segments must be perpendicular to the mirror line. If they’re both perpendicular to the mirror line, we can conclude some further information. That is, their alternate angles are equal, and they must in fact be parallel to one another. To find the second blank word in question (2), we add the line segment to the diagram. And of course, we know that 𝐶𝐶 prime is perpendicular to line 𝐿.

Finally, we consider question (3). Remember, these two triangles are congruent, which means they share angle measures. In particular, this means that the measure of angle 𝐴 must be equal to the measure of angle 𝐴 prime. Angle 𝐴 prime is 31 degrees, so angle 𝐴 is also 31 degrees.

And so, we have filled in the blanks. The correct entries were four, six, parallel, perpendicular, and 31 degrees.

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