The length of the midsegment of a trapezoid - the segment that has as its endpoints the midpoints of its legs - is half the sum of the lengths of its legs. Therefore,  Trapezoid  has as the length of its midsegment

.

Sidelengths of similar figures are in proportion. If the similarity ratio is , then the bases of Trapezoid  have length  and , so their midsegment will have length

,

meaning that the ratio of the lengths of the midsegments will be the same as the similarity ratio. Since the length of the midsegment of Trapezoid  is 91, this similarity ratio is

 .

The ratio of the length of  to that of corresponding side  is therefore , so

The similarity ratio of the trapezoids is the ratio of the length of one side to that of the corresponding side of the other. For these trapezoids, we take the ratio of the lengths of corresponding sides  and :

.

The ratio of the areas is the square of this, or 

.

The area of Trapezoid  is one half the product of the height  and the sum of bases  and :

Multiply this by the area ratio:

.

The correct response is 1,271.

The area of a rhombus is half the product of the lengths of its diagonals, so we can take advantage of this to find  from the known measurements of Rhombus :

The ratio of the area of Rhombus  to that of Rhombus is 4, so their similarity ratio is the square root of this, or 2. We can calculate  now:

The Quadrilateral  with its diagonals is shown below. We call the point of intersection :

The diagonals of a quadrilateral with two pairs of adjacent congruent sides - a kite - are perpendicular. , the diagonal that connects the common vertices of the pairs of adjacent sides, bisects the other diagonal, making  the midpoint of . Therefore, .

 also bisects the  and angles of the kite, so the result is that two 30-60-90 triangles,  and , and two 45-45-90 triangles ,  and , are formed; also, , being isosceles, is a 45-45-90 triangle. 

Examine . Since its short leg  has length 12, by the 30-60-90 Theorem, its hypotenuse, , has twice this length, or 24.

Examine . Since a leg  has length 12, by the 45-45-90 Theorem, its hypotenuse, , has length  times this, or .

Since by similarity, corresponding sides are in proportion,

Construct the perpendicular segment from  to  and let  be its point of intersection with . By construction, the trapezoid is divided into Rectangle  and right triangle . Since opposite sides of a rectangle are congruent,  and ; as a consequence of the latter, . By the Pythagrean Theorem, the length of the hypotenuse  of right triangle   can be calculated from the length of legs  and :

The figure, with the segment and the calculated measurements, is below.

Since Trapezoid  Trapezoid , by proportionality of corresponding sides of similar figures:

A rhombus has four sides the same length, so each side of Rhombus  has length one fourth of 80, or 20; each side of Rhombus  has length one fourth of 180, or 45. 

The diagonals of a rhombus are each other's perpendicular bisectors, so, if we let  be the point of intersection of the diagonals of Rhombus ,  and , we form four congruent right triangles. 

We will examine . ; and,.

By the Pythagorean Theorem, 

and 

Since corresponding sides of similar figures are in proportion, so are corresponding diagonals. Therefore,

The angles of the rhombus measure:

;

, since opposite angles of a rhombus, as in any other parallelogram, are congruent;

 and , since consecutive angles of a rhombus are supplementary (the sum of their degree measures is 180).

The diagonals of a rhombus are each other's perpendicular bisectors as well as the bisectors of the angles, so  and , whose point of intersection we will call , divide Rhombus  into four 30-60-90 triangles. If we examine , we see that its short leg  has length half that of ; , so . By the 30-60-90 Triangle Theorem, long leg  has length  this, or , and the diagonal  has measure twice this, or .

The ratio of the lengths of corresponding diagonals of the rhombuses is the same as the similarity ratios of the sides, so the similarity ratio of Rhombus  to Rhombus  is 

The ratio of the areas is the square of the similarity ratio, which is

That is, 12 to 1.

The Quadrilateral  with its diagonals is shown below. We call the point of intersection :

The diagonals of a quadrilateral with two pairs of adjacent congruent sides - a kite - are perpendicular; also,  bisects the  and angles of the kite. Consequently, two congruent 30-60-90 triangles,  and , and two congruent 45-45-90 triangles ,  and , are formed; also, , being an isosceles right triangle, is a 45-45-90 triangle. , the hypotenuse of , has length 16, so by the 30-60-90 Triangle Theorem, its shorter leg  has length half this, or 8. Also,  is a leg of , so by the 45-45-90 Theorem, the hypotenuse  has length  times this, or .

Corresponding sides of similar quadrilaterals are in proportion, so

We will assume that  and  have common measure 1 for the sake of simplcity; this reasoning is independent of the actual measure of .

The Quadrilateral  with its diagonals is shown below. We call the point of intersection :

The diagonals of a quadrilateral with two pairs of adjacent congruent sides - a kite - are perpendicular; also,  bisects the  and angles of the kite. Consequently,  is a 30-60-90 triangle and  is a 45-45-90 triangle. By the 30-60-90 Theorem, since  and  are the short leg and hypotenuse of ,

 .

By the 45-45-90 Theorem, since  and  are a leg and the hypotenuse of , 

The similarity ratio of Quadrilateral  to Quadrilateral  can be found by finding the ratio of the length of side  to corresponding side :

The ratio of the areas is the square of the similarity ratio: 

The correct choice is that Quadrilateral  has area twice that of Quadrilateral .

Let  be the longer base. Then, since this is ten centimeters (= 1 decimeter) longer than the shorter, the shorter base has length . Substitute  in the formula for the area of a trapezoid, and solve for :

 centimeters