Question Video: Using Properties of Area Formulas to Identify Shapes with Equal Areas | Nagwa Question Video: Using Properties of Area Formulas to Identify Shapes with Equal Areas | Nagwa

# Question Video: Using Properties of Area Formulas to Identify Shapes with Equal Areas Mathematics • Second Year of Preparatory School

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Given that ππ β πΎπ·, which of the following has the same area as β³πππΎ? [A] β³πΆππ» [B] β³πΆππ» [C] β³π»ππ [D] ππππ» [E] π»ππΎπΆ

04:03

### Video Transcript

Given that line ππ is parallel to line πΎπ·, which of the following has the same area as triangle πππΎ? (A) Triangle πΆππ», (B) triangle πΆππ», (C) triangle π»ππ, (D) quadrilateral ππππ», or (E) quadrilateral π»ππΎπΆ.

Letβs begin by highlighting the shape weβre looking at. Triangle πππΎ is highlighted here. Weβre looking for a shape that has an equal area. So letβs recall how to find the area of a triangle. For a triangle whose height is β units and base is π units, the area of that triangle will be equal to one-half times the base times the height. Recall that the base of a triangle doesnβt necessarily need to be at the bottom. That just depends on its orientation.

So here we can label the base as the length of line segment ππ. The height is then the perpendicular distance between line segment ππ and the line that has point πΎ on it. Weβve already been told that the line passing through ππ is parallel to the line passing through πΎπ·. This means that the perpendicular distance between the line ππ and the line πΎπ· can be labeled as β units. It doesnβt matter where on the line weβre looking. This is really useful because we now know that any triangle created here will have a height of β units. The triangle πΆππ» would have a perpendicular height of β units, as would the height of the triangle ππ·π.

Weβve shown so far that triangles πππΎ, πΆππ», and ππ·π have the same height, β units. Looking closely, we see that the bases of these triangles have been marked out with a dash mark. This means we can go further and say that these three triangles have the same base, π units. This means weβve shown that the area of these three triangles will be the same.

However, we need to go ahead and consider the other two quadrilaterals that were listed here. ππππ» Iβve highlighted here in yellow and π»ππΎπΆ in green. These quadrilaterals have one pair of parallel sides, which means theyβre trapezoids. The area of a trapezoid is equal to one-half base one plus base two times the height. While these trapezoids do share the same height as the triangles weβve already mentioned, thatβs the perpendicular distance between the lines ππ and πΎπ·, in order for one of these trapezoids to have the same area as triangle πππΎ, we would have to be able to prove that the two parallel bases are equal in length to the base ππ. And thereβs nothing on this diagram that gives us enough information or to indicate that this would be the case.

What we can be certain of is that the area of triangle πππΎ is equal to the other two triangles weβve considered. This is option (B) triangle πΆππ». With regard to the other two triangles in this list, triangle πΆππ» and triangle π»ππΆ, it is true that these two triangles would have the same height as our three triangles weβve already considered. However, we have no information about the bases of these triangles. And therefore, we cannot claim that they have the same area as triangle πππΎ. From this list, the only triangle that certainly has the same area as triangle πππΎ is triangle πΆππ».

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