Other Quadrilaterals - GMAT Quantitative

In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can check which dimensions are those of a rectangle similar to the given one by first calculating the ratio of the length to the width for the given rectangle, and then doing the same for each of the answer choices until we find which has an equal ratio between its dimensions:

So in order for a rectangle to be similar to the given rectangle, this must be the ratio of its length to its width. Now we check the answer choices, in no particular order, for one with this ratio:

We can see that only the rectangle with a length of and a width of has the same ratio as the given rectangle, so this is the similar one.

In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can calculate the ratio of length to width for the given rectangle, and then check the answer choices for the rectangle whose dimensions have the same ratio:

Now we check the answer choices, in no particular order, and the dimensions with the same ratio are those of the rectangle that is similar:

We can see that a rectangle with a length of and a width of has the same ratio of dimensions as the given rectangle, so this is the one that is similar.

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:

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

Opposite sides of a parallelogram are congruent; if , then , violating this condition.

Consecutive angles of a parallelogram are supplementary; if , then , violating this condition.

Opposite angles of a parallelogram are congruent; if , then , violating this condition.

Adjacent sides of a parallelogram, however, may or may not be congruent, so the condition that would not by itself prove that the quadrilateral is not a parallelogram.

The four interior angles of a quadrilateral measure a total of , so we test each group of numbers to see if they have this sum.

This last group does not have the correct sum, so it is the correct choice.

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary. An isosceles trapezoid has this characteristic; this can be proved by the fact that base angles are congruent, and by the Same-Side Interior Angles Statement. For a parallelogram to have this characteristic, since opposite angles are congruent also, all angles must measure ; the rectangle fits this description, but the rhombus does not.

Case 1: The two angles are opposite angles of the parallelogram. In this case, they are congruent, and

Case 2: The two angles are consecutive angles of the parallelogram. In this case, they are supplementary, and

The diagonals of a rhombus always intersect at right angles, so . The measures of the interior angles of the rhombus are irrelevant.

Two opposite angles of a quadrilateral inscribed inside a circle are supplementary, so

The four sides of a rhombus are congruent, so a diagonal of the rhombus cuts it into two isosceles triangles. The two angles adjacent to the diagonal are congruent, so the third angle, the one marked, measures:

To prove that Polygon is also a rectangle, we need to prove that any one of its angles is a right angle.

If , then by definition of perpendicular lines, is right.

If , then, since and form a linear pair, is right.

If , then, by the Converse of the Pythagorean Theorem, is a right triangle with right angle .

If and are complementary angles, then, since

, making right.

However, since, by definition of a parallelogram, , by the Alternate Interior Angles Theorem, regardless of whether the parallelogram is a rectangle or not.

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about any regular polygon, so we can eliminate those two choices as well.

The correct choice is that each figure can have a circle circumscribed about it.

As can be seen from this diagram, this is a parallelogram with base 8 and height 4:

The area of this parallelogram is the product of its base and its height:

As can be seen in this diagram, this is a trapezoid with bases 10 and 5 and height 8.

Setting in the following formula, we can calculate the area of the trapezoid:

Quadrilateral is a composite of two right triangles, and , so we find the area of each and add the areas. First, we need to find and , since the area of a right triangle is half the product of the lengths of its legs.

By the Pythagorean Theorem:

Also by the Pythagorean Theorem:

The area of is .

The area of is .

Add the areas to get , the area of Quadrilateral .

The quadrilateral formed is a trapezoid with two horizontal bases. One base connects (0,0) and (9,0) and therefore has length ; the other connects (4,7) and (7,7) and has length . The height is the vertical distance between the two bases, which is the difference of the coorindates: . Therefore, the area of the trapezoid is

The quadrilateral is a trapezoid with horizontal bases; one connects and and has length , and the other connects and and has length . The height is the vertical distance between the bases, which is the difference of the -coordinates; this is . Substitute in the formula for the area of a trapezoid: