Question Video: Solving Problems Related to a Reflected Triangle | Nagwa Question Video: Solving Problems Related to a Reflected Triangle | Nagwa

# Question Video: Solving Problems Related to a Reflected Triangle Mathematics • First Year of Preparatory School

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In the following figure, β³π΄β²π΅β²πΆβ² is the image of β³π΄π΅πΆ by reflection in the line πΏ. (1) Fill in the blanks: The length of π΄β²πΆβ² = οΌΏ cm, and the length of π΄β²π΅β² = οΌΏ cm. (2) Fill in the blanks: Line segment π΄π΄β² is οΌΏ to line segment π΅π΅β², and line segment πΆπΆβ² is οΌΏ to line πΏ. (3) Find the measure of β π΄.

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### Video Transcript

In the following figure, triangle π΄ prime π΅ prime πΆ prime is the image of triangle π΄π΅πΆ by reflection in the line πΏ. (1) Fill in the blanks. The length of π΄ prime πΆ prime equals blank centimeters, and the length of π΄ prime π΅ prime equals blank centimeters. (2) Fill in the blanks. Line segment π΄π΄ prime is blank to line segment π΅π΅ prime, and line segment πΆπΆ prime is blank to line πΏ. (3) Find the measure of angle π΄.

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 π΄πΆ is congruent to line segment π΄ prime πΆ prime. They must have the same lengths. Since line segment π΄πΆ is four centimeters, line segment π΄ prime πΆ prime must also be four centimeters. And we put four in the first blank space.

Next, line segment π΄π΅ must be congruent to line segment π΄ prime π΅ prime. And so, π΄ prime π΅ prime must be six centimeters in length. And six goes in our second blank space.

Letβs now consider question (2). First, we add line segments π΄π΄ prime and π΅π΅ prime to the diagram. We know that these line segments must be perpendicular to the mirror line. If theyβ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 πΆπΆ prime is perpendicular to line πΏ.

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 π΄ must be equal to the measure of angle π΄ prime. Angle π΄ prime is 31 degrees, so angle π΄ 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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