Question Video: Finding the Area of a Triangle between Parallel Lines | Nagwa Question Video: Finding the Area of a Triangle between Parallel Lines | Nagwa

# Question Video: Finding the Area of a Triangle between Parallel Lines Mathematics • Second Year of Preparatory School

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In the opposite figure, the line passing through the points π΄π΅ β₯ the line passing through the points πΆπ·. The line segment π΄πΆ β₯ the line segment π΅π·, and the line segment πΉπΊ β₯ the line segment π΄π΅, where πΉπΊ = 4 cm. If the area of β³π΄π΅πΈ = 12 cmΒ², find the length of the line segment πΆπ·.

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### Video Transcript

In the opposite figure, the line passing through the points π΄ and π΅ is parallel to the line passing through the points πΆ and π·. The line segment π΄πΆ is parallel to the line segment π΅π·. And the line segment πΉπΊ is perpendicular to the line segment π΄π΅, where the length πΉπΊ is equal to four centimeters. If the area of the triangle π΄π΅πΈ is equal to 12 centimeters squared, find the length of the line segment πΆπ·.

Letβs start with labeling the diagram with the information given. The length of πΉπΊ is four centimeters, and the area of the triangle π΄π΅πΈ is 12 centimeters squared. Recall that the area of a triangle is given by one-half times its base, π, times its perpendicular height, β.

Looking at the triangle π΄π΅πΈ, we can take its base length to be the length of the line segment π΄π΅. The height of the triangle is the distance between the top line and the point πΈ, which is equivalent to the distance between the top and bottom lines. Since these lines are parallel and the line segment πΉπΊ is perpendicular to them, the distance between both lines is just the length πΉπΊ. Therefore, the area of the triangle is one-half times π΄π΅ times πΉπΊ, which we are given as equal to 12 centimeters squared.

We are also given that the length πΉπΊ is equal to four centimeters. So on the left-hand side of this equation, we have one-half times π΄π΅ times four, which is equal to two π΄π΅. So we have two π΄π΅ equals 12. Dividing both sides by two gives us the length of the line segment π΄π΅, six centimeters.

Now consider the quadrilateral π΄π΅πΆπ·. Since, as given in the question, the line segments π΄π΅ and πΆπ· are parallel, the line segments π΄πΆ and π΅π· are also parallel. So this is a parallelogram, and the parallel sides are of equal length. Therefore, the length of the line segment πΆπ· is equal to the length of the line segment π΄π΅, which is six centimeters.

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