GMAT Math : DSQ: Calculating whether quadrilaterals are similar

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

True or false: Rhombus $ABCD \sim$ Rhombus EFGH .

Statement 1: $\angle A \cong \angle E$

Statement 2: $\frac{AC}{EG} = \frac{BD}{FH}$

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. A rhombus being a parallelogram, its opposite angles and its adjacent angles are supplementary. From this fact and Statement 1 alone, it follows that

$\angle A \cong \angle E$ , $\angle B \cong \angle F$ , $\angle C\cong \angle G$ , and $\angle D \cong \angle H$ .

By definition of a rhombus, all of its sides are congruent. By substitution,

$\frac{AB}{EF} = \frac{BC}{FG}= \frac{CD}{GH} = \frac{DA}{HE}$ .

All side proportions hold as well as all angle congruences, so the similarity statement holds.

Assume Statement 2 alone. Construct the diagonals of the rhombuses, as follows:

Rhombuses

In each rhombus, the diagonals are each other's perpendicular bisector. If

$\frac{AC}{EG} = \frac{BD}{FH}$

then

$\frac{AC \div 2}{EG \div 2} = \frac{BD \div 2}{FH \div 2}$

$\frac{AM}{EN} = \frac{BM}{FN}$

Since $\angle AMB \cong \angle ENF$ , both angles being right, it follows via the Side-Angle-Side Smiilarity Theorem that

$\bigtriangleup MAB \sim \bigtriangleup NEF$ ,

and , by similarity,

$m \angle MAB = m \angle NEF$ .

By a similar argument,

$m \angle MAD = m \angle NEH$ ,

and by angle addition,

$m \angle BAD = m \angle FEH$ .

As with Statement 1 alone, congruence of one set of corresponding angles in two rhombuses leads to the similarity of the two.

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

What is the perimeter of Rhombus ABCD ?

Statement 1: AC = 6

Statement 2: BD = 8

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Each statement gives the length of one diagonal of the rhombus. Knowing one diagonal is not enough to give the perimeter of the rhombus.

Knowing the lengths of both diagonals, which is the case if both statements are assumed, is enough to determine the perimeter. The rhombus in question, along with its diagonals, is as shown below:

Rhombus

As marked in the diagram, the diagonals are perpendicular, and they are also are each other's bisector. It follows from the given lengths that AM = 3 and BM = 4 , so can be calculated using the Pythagorean Theorem. The perimeter is four times , since all sides of a rhombus are congruent.

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

Isosceles Trapezoid TRAP has bases $\overline{TR}$ and $\overline{AP}$ .

Isosceles Trapezoid USBQ has bases $\overline{US}$ and $\overline{BQ}$ .

True or false:

Trapezoid $TRAP \sim$ Trapezoid USBQ

Statement 1: $\overline{TP} \cong \overline{UQ}$

Statement 2: $\overline{RA} \cong \overline{SB}$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

Assume both statements are true. We show that they provide insufficient information by examining two scenarios.

If Trapezoid $TRAP \cong$ Trapezoid USBQ , then $\overline{TP} \cong \overline{UQ}$ and $\overline{RA} \cong \overline{SB}$ , so the conditions of both statements are met; also, since the trapezoids are congruent, they are also similar.

Now examine isosceles trapezoid TRAP below, in which , , , and are positioned on the bases so that $\overline{TP} || \overline{UQ}$ and $\overline{RA} || \overline{SB}$ .

Trapezoid_2

Since $\overline{TP} || \overline{UQ}$ and $\overline{TU} || \overline{PQ}$ , Quadrilateral TUPQ is a parallelogram, and $\overline{TP} \cong \overline{UQ}$ ; similarly, $\overline{RA} \cong \overline{SB}$ . Therefore, Trapezoid USBQ is also isosceles, and the conditions of both statements are met. However, corresponding sides are not in proportion, since $\overline{TP} \cong \overline{UQ}$ , but $\overline{TR} \ncong \overline{US}$ ; consequently, the trapezoids are not similar.

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

True or false: Rhombus $RHOM \sim$ Rhombus SIPN .

Statement 1: $OM \cdot 7 = PN$ and $HO \cdot 7 = IP$

Statement 2: The area of Rhombus SIPN is 49 times that of Rhombus RHOM .

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that corresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary—that is, their angle measures total $180^{\circ }$ . Therefore, it is necessary and sufficient to prove just one pair of corresponding angles congruent.

The two statements together give no information about the measures of any of the angles of the rhombus. Therefore, together, they do not answer the question of whether they are similar or not.

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

True or false: Rhombus $RHOM \sim$ Rhombus SIPN .

Statement 1: $\angle R$ and $\angle I$ are both $60^{\circ }$ angles.

Statement 2: The area of Rhombus RHOM is four times that of Rhombus SIPN .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that coresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary - that is, their angle measures total $180^{\circ }$ . Therefore, it is neccessary and sufficient to prove just one pair of corresponding angles congruent.

Assume Statement 1 alone. $\angle R$ is a $60^{\circ }$ angle, so any angle consecutive to it, which includes $\angle H$ , is supplementary to it—that is, the angle measures total $180^{\circ }$ . This makes $\angle H$ a $120^{\circ }$ angle. Its corresponding angle in Rhombus SIPN is $\angle I$ , which is a $60^{\circ }$ angle. Since $\angle H$ and $\angle I$ are noncongruent, it follows that Rhombus $RHOM \nsim$ Rhombus SIPN . (Note that it can be demonstrated that the rhombuses are similar, but the correct statement is Rhombus $RHOM \nsim$ Rhombus IPNS .)

Statement 2 alone provides no useful information; the relationship between the areas of the rhombuses is irrelevant.

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

True or false: Rhombus $RHOM \sim$ Rhombus SIPN .

Statement 1: $\angle R \cong \angle I$

Statement 2: $\angle R \cong \angle M$

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that coresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary—that is, their angle measures total $180^{\circ }$ . Therefore, it is neccessary and sufficient to prove just one pair of corresponding angles congruent.

Assume Statement 1 alone. $\angle R$ and $\angle I$ are not corresponding angles, so their congruence does not prove the rhombuses are similar; however it does not prove they are not similar either; for example, two squares are both rhombuses with four right angles, so they are similar and fit this condition.

Assume Statement 2 alone. $\angle R$ and $\angle M$ are two angles of the same rhombus, so similarity cannot be proved or disproved without information about the other rhombus.

Now assume both statements. $\angle R$ and $\angle M$ are consecutive angles of Rhombus RHOM ; a rhombus being a parallelogram, the degree measures of the angles total $180^{\circ }$ . From Statement 2, they are congruent, so each measures $90 ^{\circ }$ . Since $\angle R \cong \angle I$ from Statement 1, $\angle I$ also measures $90 ^{\circ }$ . Since each parallelogram has at least one right angle, each has four right angles. Therefore, corresponding angles are congruent, so the rhombuses are similar.

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

Isosceles Trapezoid TRAP has bases $\overline{TR}$ and $\overline{AP}$ .

Isosceles Trapezoid USBQ has bases $\overline{US}$ and $\overline{BQ}$ .

True or false:

Trapezoid $TRAP \sim$ Trapezoid USBQ

Statement 1: $\angle T$ and $\angle S$ are both $120^{\circ}$ angles.

Statement 2: PT=TR= RA and QU=US= SB

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent and that their corresponding sides are in proportion.

We show that Statement 1 alone provides insufficient information by examining two cases.

Case 1: Trapezoid $TRAP \cong$ Trapezoid USBQ , with $\angle T$ a $120^{\circ}$ angle.

A pair of base angles of an isosceles trapezoid are congruent, so, since $\angle T$ measures $120^{\circ}$ , so does $\angle R$ . The leg angles of a trapezoid have measures whose sum is $1 80^{\circ }$ , so $\angle A$ and $\angle P$ measure $60^{\circ}$ . By congruence of corresponding angles, $\angle S \cong \angle R$ , so $\angle S$ measures $120^{\circ}$ , and the conditions of Statement 1 are met. Furthermore, two congruent triangles are also similar, so Trapezoid $TRAP \sim$ Trapezoid USBQ .

Case 2: Examine the figure below, in which TR = US . Note that these congruent upper bases have been superimposed upon each other:

Trapezoid_3

The trapezoids are isosceles since their base angles are congruent; also, the conditions of Statement 1 are met. However, TR = US , but $BQ \ne AP$ . The sides are not in proportion, so the trapezoids are not similar.

Assume Statement 2 alone. Two congruent trapezoids whose side lengths fit the conditions of the statement are also similar. But examine this diagram, in which TR = US , PT=TR= RA , and QU=US= SB . Note that the congruent upper bases have been superimposed upon each other:

Trapezoid_4

The conditions of Statement 2 are met, but $\angle T \ncong \angle U$ , so the trapezoids are not similar.

Now assume both statements to be true. As stated before, it follows from Statement 1 that all corresponding angles are congruent. PT=TR= RA and QU=US= SB , so it follows from the Division Property of Equality that

$\frac{PT}{QU}=\frac{TR}{US}= \frac{RA}{SB}$

It remains to be demonstrated that $\frac{AP}{BQ}$ is equal to the above ratios as well. If the diagonals $\overline{RP }$ and $\overline{SQ}$ are constructed, $\bigtriangleup TRP$ and $\bigtriangleup USQ$ are formed; since $\frac{PT}{QU}=\frac{TR}{US}$ and $\angle T \cong \angle U$ , by the Side-Angle-Side Similarity Theorem, $\bigtriangleup TRP \sim \bigtriangleup USQ$ . It follows that $\angle TRP \cong \angle USQ$ , and by angle addition, since $\angle TRA \cong \angle USB$ , it follows that $\angle PRA \cong \angle QSB$ . From the Angle-Angle Postulate, the other two triangles formed are similar—that is, $\bigtriangleup RAP \sim \bigtriangleup SBQ$ —and it follows that $\frac{AP}{BQ}= \frac{RA}{SB}$ . Therefore, all four sides of the trapezoids are in proportion, and the trapezoids are similar.

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

Isosceles Trapezoid TRAP has bases $\overline{TR}$ and $\overline{AP}$ .

Isosceles Trapezoid USBQ has bases $\overline{US}$ and $\overline{BQ}$ .

True or false:

Trapezoid $TRAP \sim$ Trapezoid USBQ

Statement 1: $\angle T \cong \angle U$

Statement 2: $\angle A \cong \angle B$

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To prove two figures similar, we must prove that their corresponding angles are congruent and that their corresponding sides are in proportion.

Assume both statements are true. We show that they provide insufficient information by examining two scenarios.

Case 1: Trapezoid $TRAP \cong$ Trapezoid USBQ , then $\angle T \cong \angle U$ and $\angle A \cong \angle B$ , so the conditions of both statements are met; also, since the trapezoids are congruent, they are also similar.

Now examine isosceles trapezoid TRAP below, in which , , , and are positioned on the bases so that $\overline{TP} || \overline{UQ}$ and $\overline{RA} || \overline{SB}$ .

Trapezoid_2

Since $\overline{TP} || \overline{UQ}$ and $\overline{TU} || \overline{PQ}$ , Quadrilateral TUPQ is a parallelogram, and $\overline{TP} \cong \overline{UQ}$ ; similarly, $\overline{RA} \cong \overline{SB}$ . Therefore, Trapezoid USBQ is also isosceles. Also, by the Corresponding Angles Theorem, $\angle T \cong \angle SUQ$ and $\angle A \cong \angle SBQ$ , and the conditions of both statements are met. However, corresponding sides are not in proportion, since $\overline{TP} \cong \overline{UQ}$ , but $\overline{TR} \ncong \overline{US}$ ; consequently, the trapezoids are not similar.

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Example Question #51 : Quadrilaterals

Determine if rectangles BPSD and KMDS are similar.

I) BPSD has a perimeter of 16 units and side is 3 units long.

II) KMDS has area of 44 units and side is 6 units long.

Possible Answers:

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

Similar rectangles (or any shape for that matter) are the same shape but can be different sizes. What that means is that their side lengths all follow a common ratio; if one pair of corresponding sides follow the ratio 2:1, then all corresponding sides must follow the same ratio.

Statement I gives us the perimeter and one side of BPSD .

Statement II gives us the area and one side of KMDS .

We can find all the sides of both rectangles, but we need both statements to do so. Once we have all side lengths, we can compare them to see if they follow the same ratios.

IF BPSD has perimeter of 16 and one side is 3, we can find the other side using the following:

$16=2l+2\cdot 3$

l=5

If KMDS has area of 44 and side of 6, the other side can be found via the following:

$44=6\cdot w$

$w=\frac{44}{6}=\frac{22}{3}$

Compare the ratios of the sides to find out whether the two rectangles are similar:

$\frac{3}{6}\not\equiv \frac{5}{\frac{22}3}$

Therefore, the rectangles are not similar.

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GMAT Math : DSQ: Calculating whether quadrilaterals are similar

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

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Example Question #51 : Quadrilaterals

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