Trapezoid and rhombus relationship questions

What is the difference between a trapezoid and a rhombus? | Socratic

trapezoid and rhombus relationship questions

The three special parallelograms — rhombus, rectangle, and square — are so- called because they're special cases of the parallelogram. (In addition, the. GMAT Math Help» Data-Sufficiency Questions» Geometry Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles. The quadrilateral family tree in the figure below shows you the relationships among always, sometimes, never problems because you can use the quadrilaterals' (like from a parallelogram to an isosceles trapezoid), the answer is never.

So these two sides are parallel. And then you could make the exact same argument for the other two sides. This line up here forms a degree angle with this side. And so does this side. It forms a degree angle with this line right over here. They form the same angle with this line. So this side is parallel to that side right over there.

So this is definitely also a parallelogram. Next, we ask about a trapezoid. Now, trapezoid is interesting.

trapezoid and rhombus relationship questions

Sometimes a trapezoid is defined as any quadrilateral having at least one pair of parallel sides. Sometimes it's defined as having only one pair of parallel sides. So let me write this down.

trapezoid and rhombus relationship questions

Trapezoid, there's a debate here. It's not completely settled. Some people say at least one pair of parallel sides. That's one definition, one possible definition.

trapezoid and rhombus relationship questions

The other one is at exactly one pair of parallel sides. How we answer this question depends on which definition for trapezoid we pick. Now, the one that people most refer to is actually this one right over here, exactly one pair of parallel sides.

So when you think of a trapezoid, they think of something like this, where this side over here is parallel to that side over here and those two are not parallel.

But sometimes you'll also see this at least one pair of parallel sides. And so this would include parallelograms. It would be inclusive of parallelograms because parallelograms have two pairs of parallel sides.

BBC Bitesize - KS3 Maths - 2D and 3D shapes - Revision 2

But I'm going to go with this definition right over here, exactly one pair of parallel sides. This has two pairs of parallel sides so I will not call it a trapezoid.

trapezoid and rhombus relationship questions

But it's always important to clarify what people are talking about because some people might say a trapezoid is at least one pair of parallel sides. And if we used that definition, then we would call it a trapezoid.

So it really depends on the definition that you're using. Now, let's go on to rhombus. So a rhombus is a quadrilateral where four of the sides are congruent. So a rhombus will look like this.

All four sides have the same length. Now, the trapezoid is clearly less than that, but let's just go with the thought experiment. Now, what would happen if we went with 2 times 3? Well, now we'd be finding the area of a rectangle that has a width of 2 and a height of 3.

So you could imagine that being this rectangle right over here. So that is this rectangle right over here. So that's the 2 times 3 rectangle.

Quadrilateral properties

Now, it looks like the area of the trapezoid should be in between these two numbers. Maybe it should be exactly halfway in between, because when you look at the area difference between the two rectangles-- and let me color that in. So this is the area difference on the left-hand side. And this is the area difference on the right-hand side.

trapezoid and rhombus relationship questions

If we focus on the trapezoid, you see that if we start with the yellow, the smaller rectangle, it reclaims half of the area, half of the difference between the smaller rectangle and the larger one on the left-hand side. It gets exactly half of it on the left-hand side. And it gets half the difference between the smaller and the larger on the right-hand side.

So it completely makes sense that the area of the trapezoid, this entire area right over here, should really just be the average. It should exactly be halfway between the areas of the smaller rectangle and the larger rectangle.

So let's take the average of those two numbers.

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It's going to be 6 times 3 plus 2 times 3, all of that over 2. So when you think about an area of a trapezoid, you look at the two bases, the long base and the short base. Multiply each of those times the height, and then you could take the average of them. Or you could also think of it as this is the same thing as 6 plus 2. And I'm just factoring out a 3 here. These are all different ways to think about it-- 6 plus 2 over 2, and then that times 3.

So you could view it as the average of the smaller and larger rectangle. So you multiply each of the bases times the height and then take the average.