# Show that the diagonals of a trapezoid divide it into four triangles of which two have the same area? "equi-extension"

In general a trapezoid is defined as a quadrilateral with two parallel sides. An isosceles trapezoid is a special case of trapezoid which has lateral symmetry, meaning that one side would be a mirror of the other.

The image above depicts an isosceles trapezoid,

The area of a triangle can be found using the identity:

Where

If we consider side

Now we can also see that triangle

If we subtract the area of AED from both ABD and DCA, we can see that the areas of ABE and DCE are indeed equal.

Although I illustrated this proof with an isosceles trapezoid, none of the work was specific to a regular trapezoid. Therefore, the side triangles for any trapezoid can be shown to be equal using the same reasoning.

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To show that the diagonals of a trapezoid divide it into four triangles, two of which have the same area, we can use the concept of equi-extension.

Let's consider a trapezoid with bases ( b_1 ) and ( b_2 ) and height ( h ). We draw its diagonals, dividing the trapezoid into four triangles: ( \triangle ADC ), ( \triangle ABC ), ( \triangle ABD ), and ( \triangle BCD ).

Now, let's denote the length of the diagonals as ( d_1 ) and ( d_2 ).

Using the concept of equi-extension, we can observe that the triangles ( \triangle ABD ) and ( \triangle BCD ) have the same area because they share the same base (segment ( BD )) and have equal heights.

Similarly, triangles ( \triangle ADC ) and ( \triangle ABC ) have the same area because they share the same base (segment ( AC )) and have equal heights.

Therefore, the diagonals of a trapezoid divide it into four triangles, two of which ( ( \triangle ABD ) and ( \triangle BCD ), ( \triangle ADC ) and ( \triangle ABC )) have the same area.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you prove this theorem on trapezoids and its median? The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
- Two rhombuses have sides with lengths of #2 #. If one rhombus has a corner with an angle of #pi/3 # and the other has a corner with an angle of #(5pi)/12 #, what is the difference between the areas of the rhombuses?
- What is the area of a rhombus?
- Two rhombuses have sides with lengths of #4 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(5pi)/8 #, what is the difference between the areas of the rhombuses?
- A parallelogram has sides with lengths of #9 # and #8 #. If the parallelogram's area is #63 #, what is the length of its longest diagonal?

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