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 𝑀𝑋 β€– 𝐾𝐷, which of the following has the same area as △𝑁𝑀𝐾? [A] △𝐢𝑁𝐻 [B] △𝐢𝑍𝐻 [C] △𝐻𝑁𝑍 [D] 𝑍𝑂𝑋𝐻 [E] 𝐻𝑁𝐾𝐢

04:03

Video Transcript

Given that line 𝑀𝑋 is parallel to line 𝐾𝐷, which of the following has the same area as triangle 𝑁𝑀𝐾? (A) Triangle 𝐢𝑁𝐻, (B) triangle 𝐢𝑍𝐻, (C) triangle 𝐻𝑁𝑍, (D) quadrilateral 𝑍𝑂𝑋𝐻, or (E) quadrilateral 𝐻𝑁𝐾𝐢.

Let’s begin by highlighting the shape we’re looking at. Triangle 𝑁𝑀𝐾 is highlighted here. We’re looking for a shape that has an equal area. So let’s recall how to find the area of a triangle. For a triangle whose height is β„Ž units and base is 𝑏 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 doesn’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 𝑀𝑁. The height is then the perpendicular distance between line segment 𝑀𝑁 and the line that has point 𝐾 on it. We’ve already been told that the line passing through 𝑀𝑋 is parallel to the line passing through 𝐾𝐷. This means that the perpendicular distance between the line 𝑀𝑋 and the line 𝐾𝐷 can be labeled as β„Ž units. It doesn’t matter where on the line we’re looking. This is really useful because we now know that any triangle created here will have a height of β„Ž units. The triangle 𝐢𝑍𝐻 would have a perpendicular height of β„Ž units, as would the height of the triangle 𝑋𝐷𝑂.

We’ve shown so far that triangles 𝑁𝑀𝐾, 𝐢𝑍𝐻, and 𝑋𝐷𝑂 have the same height, β„Ž 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, 𝑏 units. This means we’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. 𝑍𝑂𝑋𝐻 I’ve highlighted here in yellow and 𝐻𝑁𝐾𝐢 in green. These quadrilaterals have one pair of parallel sides, which means they’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 we’ve already mentioned, that’s the perpendicular distance between the lines 𝑀𝑋 and 𝐾𝐷, in order for one of these trapezoids to have the same area as triangle 𝑁𝑀𝐾, we would have to be able to prove that the two parallel bases are equal in length to the base 𝑀𝑁. And there’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 𝑁𝑀𝐾 is equal to the other two triangles we’ve considered. This is option (B) triangle 𝐢𝑍𝐻. With regard to the other two triangles in this list, triangle 𝐢𝑁𝐻 and triangle 𝐻𝑁𝐢, it is true that these two triangles would have the same height as our three triangles we’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 𝑁𝑀𝐾. From this list, the only triangle that certainly has the same area as triangle 𝑁𝑀𝐾 is triangle 𝐢𝑍𝐻.

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