Question Video: Proving Equality of the Area of Triangles between Parallel Lines | Nagwa Question Video: Proving Equality of the Area of Triangles between Parallel Lines | Nagwa

# Question Video: Proving Equality of the Area of Triangles between Parallel Lines Mathematics • Second Year of Preparatory School

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Given that line π΅π is parallel to line π΄π, which of the following has the same area as β³π΄π΅πΆ? [A] β³πΏπΈπ [B] β³πΏππ [C] π΅πΆπΈπ· [D] π·πΈππΏ [E] π·πΉππΏ

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### Video Transcript

Given that line π΅π is parallel to line π΄π, which of the following has the same area as triangle π΄π΅πΆ? (A) Triangle πΏπΈπ, (B) triangle πΏππ, (C) π΅πΆπΈπ·, (D) π·πΈππΏ, or (E) π·πΉππΏ.

First of all, letβs go ahead and identify triangle π΄π΅πΆ. Thatβs this triangle. And we can also mark that line π΅π is parallel to line π΄π. There are few other things we should consider, the first being how we find the area of a triangle. The area equals one-half times the height times the base. If we say that the distance from π΄ to πΆ is the base π΅, we notice that this is the same distance from πΈ to πΉ and from πΏ to π.

We also need to note that the height of a triangle is the perpendicular distance from the base to the opposite vertex. Between these two parallel lines, we have three triangles that all have the same base. And because we know that these two lines are parallel, the distance between the base and the opposite vertex for all three of these triangles will be the same. And so we can say the area of triangle π΄π΅πΆ is equal to the area of triangle πΈπ·πΉ, which is equal to the area of triangle πΏππ. And triangle πΏππ is one of our answer choices.

If we consider triangle πΏπΈπ, we see that this triangle does share the same height as triangle π΄π΅πΆ. However, it has a larger base, which means it will have a slightly larger area. And just a note about the three quadrilaterals we have as answer choices, all three of these are trapezoids. And when we find the area of a trapezoid, we take base one, add base two, divide by two, and then multiply by the height.

Now, these trapezoids that are listed do share the height of our triangle π΄π΅πΆ. However, we donβt have enough information to find base one and base two. And therefore, we cannot say that these trapezoids would have the same area as triangle πΏππ.

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