Calculating whether quadrilaterals are similar - GMAT Quantitative

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

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.

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

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:

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 .

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,

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

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.

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

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: