Question Video: Constructing a Triangle and an Angle Bisector | Nagwa Question Video: Constructing a Triangle and an Angle Bisector | Nagwa

# Question Video: Constructing a Triangle and an Angle Bisector Mathematics • First Year of Preparatory School

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Given that π΄π΅πΆ is a triangle, use a ruler and a pair of compasses to draw the triangle and bisect β πΆ by the bisector the ray πΆπ· that intersects the ray π΄π΅ at π·. Use the ruler to measure the length of the line segment π΄π· to the nearest one decimal place.

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### Video Transcript

Given that π΄π΅πΆ is a triangle, use a ruler and a pair of compasses to draw the triangle and bisect angle πΆ by the bisector the ray from πΆ through π· that intersects the ray from π΄ through π΅ at π·. Use the ruler to measure the length of the line segment π΄π· to the nearest one decimal place. Option (A) 4.2 centimeters, option (B) 3.2 centimeters, option (C) 4.7 centimeters, option (D) 3.7 centimeters. Or is it option (E) 5.2 centimeters?

In this question, we are asked to draw a triangle of given lengths using a ruler and a pair of compasses, bisect the angle at πΆ in a way to find a point π·, and then measure the length of the line segment π΄π· to the nearest tenth of a centimeter.

Letβs start with the first part of the question. We are asked to draw the triangle with the given lengths. We might be tempted to skip this step and use the given diagram. However, these are not always accurate. So it is a good idea to construct the triangle ourselves, although in this case the diagram is accurate.

To construct a triangle with a compass and ruler, we need to start by constructing one side of the triangle. Letβs say we start with π΅πΆ, which is of length five centimeters. We can then trace a circle of radius seven centimeters centered at πΆ and a circle of radius eight centimeters centered at π΅ and label either point of intersection between the circles π΄. We can then use the radius of each circle to note that triangle π΄π΅πΆ has the desired side lengths.

The second part of this question asks us to bisect the angle at πΆ, that is, the internal angle of the triangle. We can recall that the bisector will split this angle into two congruent angles. We can also note that this bisector will lie inside the triangle, so point π· will lie on the line segment π΄π΅. We do not need to extend this side. We then recall that we can bisect angles by a construction. First, we trace a circle centered at the vertex of the angle we want to bisect and label the points of intersection between the circle and the sides π΄ prime, π΅ prime as shown. We can then sketch congruent circles centered at π΄ prime and π΅ prime that intersect at a point on the same side as the angle we want to bisect. We will call this point of intersection πΈ.

We can then conclude that the line between πΆ and πΈ bisects the angle at πΆ. We can extend this bisector to intersect the opposite side and label the point of intersection π· as shown. We can then measure the length of the line segment π΄π· with our ruler. And if we do, we get 4.7 centimeters to the nearest tenth of a centimeter, which is option (C).

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