# GMAT Math : Calculating whether quadrilaterals are similar

## Example Questions

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### Example Question #1 : Calculating Whether Quadrilaterals Are Similar

Which of the following rectangles is similar to one with a length of    and a width of   ?

Explanation:

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.

### Example Question #2 : Calculating Whether Quadrilaterals Are Similar

Which of the following dimensions would a rectangle need to have in order to be similar to one with a length of    and a width of   ?

Explanation:

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.

### Example Question #3 : Calculating Whether Quadrilaterals Are Similar

Refer to the above Trapezoid . There exists Trapezoid  such that

Trapezoid  Trapezoid , and the length of the midsegment of Trapezoid  is 91.

Give the length of .

Explanation:

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

### Example Question #4 : Calculating Whether Quadrilaterals Are Similar

Refer to the above Trapezoid . There exists Trapezoid  such that

Trapezoid  Trapezoid , and  has length 66.

To the nearest whole, give the area of Trapezoid .

Explanation:

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.

### Example Question #5 : Calculating Whether Quadrilaterals Are Similar

Rhombus  Rhombus ; Rhombus  has area 90; Rhombus  has area 360. What is the length of diagonal  ?

Explanation:

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:

### Example Question #6 : Calculating Whether Quadrilaterals Are Similar

In Quadrilateral ,  ,  is a right angle, and diagonal  has length 24.

Give the length of .

Explanation:

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,

### Example Question #31 : Other Quadrilaterals

Refer to the above Trapezoid . There exists Trapezoid  such that

Trapezoid  Trapezoid , and  has length 60.

Give the length of .

Explanation:

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:

### Example Question #8 : Calculating Whether Quadrilaterals Are Similar

Rhombus  Rhombus . Rhombus  has perimeter 80; Rhombus  has perimeter 180; .

Find the length of diagonal .

Explanation:

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,

### Example Question #9 : Calculating Whether Quadrilaterals Are Similar

Rhombus  Rhombus .

Give the ratio of the area of Rhombus  to that of Rhombus .

Explanation:

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.

### Example Question #10 : Calculating Whether Quadrilaterals Are Similar

In Quadrilateral ,  , and  is a right angle; .

Give the length of .

Explanation:

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

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