Other Quadrilaterals - GMAT Quantitative Reasoning

A circle can be circumscribed about a quadrilateral if and only if both pairs of opposite angles are supplementary. This is not proved or disproved by Statement 1 alone:

Case 1:

This is not a rectangle, and opposite angles are supplementary, so a circle can be constructed to circumscribe the quadrilateral.

Case 2:

This is not a rectangle, and opposite angles are not supplementary, so a circle cannot be constructed to circumscribe the quadrilateral.

From Statement 2, however, it follows that a two opposite angles are not a supplementary pair, so a circle cannot be circumscribed about it.

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. Assume without loss of generality that and are the pairs of base angles.

Then, since base angles are congruent, and . Since the bases of a trapezoid are parallel, from the Same-Side Interior Angles Theorem, and are supplementary, and, subsequently, so are and , as well as and .

If , then and form a supplementary pair, as their measures total ; since the measures of the angles of a quadrilateral total , the measures of and also total , making them supplementary as well.

Therefore, it follows from either statement that both pairs of opposite angles are supplementary, and that a circle can be circumscribed about the quadrilateral.

and , the diagonals of Parallelogram , are perpendicular if and only if Parallelogram is also a rhombus.

Opposite sides of a parallelogram are congruent, so if Statement 1 is assumed, . Parallelogram a rhombus; subsequently, .

The angle measures are irrelevant, so Statement 2 is unhelpful.

For the diagonals of a quadrilateral to be perpendicular, the quadrilateral must be a kite or a rhombus - in either case, there must be two pairs of adjacent congruent sides. Neither statement alone proves this, but both statements together do.

From Statement 1 alone, we can calculate , since two opposite angles of a quadrilateral inscribed inside a circle are supplementary:

From Statement 2 alone, we can calculate , since the degree measure of an inscribed angle of a circle is half that of the arc it intersects:

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

If we assume Statement 1 alone, that , then, since and form a linear pair, is right.

If we assume Statement 2 alone, that , it follows from the converse of the Pythagorean Theorem that is a right triangle with right angle .

Either way, we have proved that the parallelogram is a rectangle.

It is necessary and sufficient to prove that one of the angles of the parallelogram is a right angle.

Assume Statement 1 alone - that . and are supplementary, since they are same-side interior angles of parallel lines. Since , is also supplementary to . But as corresponding angles of parallel lines, . Two angles that are conruent and supplementary are both right angles, so is a right angle.

Assume Statement 2 alone - that and are complementary angles, or, equivalently, . Since the angles of a triangle have measures that add up to , the third angle of , which is , measures , and is a right angle.

Either statement alone proves a right angle and subsequently proves a rectangle.

Assume Statement 1 alone. By congruence, and , making Quadrilateral a parallelogram. However, no clue is given to whether any angles are right or not, so whether the quadrilateral is a rectangle or not remains open.

Assume Statement 2 alone. By congruence, opposite sides , but no clue is provided as to the lengths of opposite sides and . Also, , but no clue is provided as to whether the angles are right. A rectangle would have both characteristics, but so would an isosceles trapezoid with legs and .

Assume both statements are true. Quadrilateral is a parallelogram as a consequence of Statement 1. Since and are consecutive angles of the parallelogram, they are supplementary, but they are also congruent as a consequence of Statement 2. Therefore, they are right angles, and a parallelogram with right angles is a rectangle.

Statement 1 alone is insufficient to answer the question; A quadrilateral in which and are right angles, , and fits the statement, as well as a rectangle, which by defintion has four right angles.

Statement 2 alone is insufficient as well, as a parallelogram with acute and obtuse angles, as well as a rectangle, fits the description.

Assume both statements, and construct diagonal to form two triangles and . By Statement 1, both triangles are right with congruent legs , and congruent hypotenuses, both being the same segment . By the Hypotenuse Leg Theorem, . By congruence, . The quadrilateral, having two sets of congruent opposite sides, is a parallelogram; a parallelogram with right angles is a rectangle.

To find the diagonal of a rectangle, we need the length of two sides.

I) Gives us the length of one side.

II) Lets us find the length of the next side.

Use I) and II) with Pythagorean Theorem to find the diagonal.

Conversely, recognize that we are making a 3/4/5 Pythagorean Triple and see that the last side is 30 fathoms.

Statement 1: The perimeter of the square is known.

A square has four equal sides. Write the perimeter formula for squares.

The side length of a square is a fourth of the perimeter.

Statement 2: A secondary square with a known area encloses the primary square where all 4 corners of the primary square touch each of secondary square's edges.

If the primary square corners touch each of the secondary square edges, they must touch at the midpoint of each edge. Since Statement 2 mentions that the secondary square area is known, it is possible to solve for the edge length and the diagonal of the secondary square. Write the formula for the area of a square.

The diagonal of the secondary square can be solved by using the Pythagorean Theorem.

The side length of the secondary square also must equal the diagonal of the primary square.

Therefore:

Each of the statements, 1 and 2, provide something integral to calculating the length of a diagonal: an angle and the length of the sides connected to its vertex. The diagonal would be the third leg of the resulting triangle, and can be calculated using the law of cosines:

Since this is a parellelogram, knowing one angle allows us to know all the angles.

Area alone is not enough information. Imagine, for instance, a parallelogram with a shorter side of and a longer side of . The diagonal would be well above

However, with the perimeter, the smaller and larger sides must add up to one half of it, .

The longer diagonal reaches its maximum with the larger internal angle widens towards degrees, and the parallelogram flattens into a line. Using the law of cosines, this translates to:

.

At its max, the diagonal could be no greater than

Statement 1: The area is 1.

Given the area of a square is sufficient to find the diagonal of the square. The formula to find the side length of the square is:

The Pythagorean theorem can then be used to solve for the length of the diagonal.

Statement 2: The square is inscribed inside of a circle with all 4 corners touching the edge of the circle, and the circle has an area of 1.

This statement is also sufficient to solve for the diagonal. With all 4 corners of the square touching the circle, the diagonal of the square is simply the length from one of the square's opposing 90 degree angles to its opposite angle, which is the diameter of the circle.

Given the area of the circle, use the formula to determine the diameter of the circle, which is also the diagonal of the square.

Therefore:

With a ratio of sides of 2:3, statement 1 tells us PS and QR are 9 units long. Knowing the ratio we can determine both PQ and SR are 6. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ: . Therefore, statement 1 alone is sufficient.

With a ratio of sides of 2:3, statement 2 tells us PQ and SR are 6 units long. Knowing the ratio we can determine both PS and QR are 9 units long. Since rectangles have four 90 degree angles, we can use the Pythagorean Theorem to solve for the length of SQ: . Therefore, statement 2 alone is sufficient.

Therefore, the correct answer is EACH statement ALONE is sufficient.

By definition, a parallelogram is a quadrilateral with 2 sets of parallel sides and diagonals that bisect each other.

From statement 1 we cannot determine the length of AC, since ABCD is only defined as a parallelogram and not necessarily a rectangle (which is a special parallelogram with equal diagonals). Therefore, statement 1 alone is not sufficient.

From statement 2, we can determine that since the definition of a parallelogram states the diagonals bisect each other. Therefore, Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

With the information from statement 1, we don't know the measurements of any parts of PQRS. Therefore, Statement 1 alone is not sufficient.

With the information from statement 2, we know that , but since we have no confirmed definition of the shape of PQRS, we can't extrapolate the length of SQ to any other length. Therefore, statement 2 alone is not sufficient.

Even if we look at both statements 1 and 2 together, quadrilateral PQRS looks like a rectangle, but we are only told it is a parallelogram. The figures on the GMAT are not drawn to scale, therefore your eyes cannot be trusted. No difinitive length of PR can be determined. Therefore, statements 1 and 2 together are not sufficient.

Therefore, the correct answer is Statements (1) and (2) TOGETHER are NOT sufficient.

Find the length of the diagonal of rectangle DEFT

I) Side DE is 3 times the length of side TD

II) The perimeter of the rectangle is

We need to find the diagonal of a rectangle. If we work back from there, we can see that to find the diagonal, we will need to the lengths of both sides of the rectangle.

I) Gives us the relationship between the two sets of sides:

II) Tells us the perimeter, for which we can write and equation:

Now, use the clue from I) to simplify our equation in II) to one variable:

So now that we have our side lengths, use Pythagorean Theorem to find the diagonal:

Knowing one pair of sides of a quadrilateral to be congruent is not alone sufficient to prove that figure to be a parallelogram, nor is knowing one pair of sides of a quadrilateral to be parallel. But knowing both about the same pair of sides is sufficient by a theorem of parallelograms.