Calculating the length of the diagonal of a quadrilateral - GMAT Quantitative

Write the formula for finding the area of a square given the diagonal.

Rearrange the equation so that the diagonal term is isolated.

Substitute the known area and simplify.

The diagonal of a square is simply the hypotenuse of a right triangle whose other two sides are the length and width of the square. Because all sides of a square are equal in length, this means the length and width are both , which gives us a right triangle with a base of and a height of , for which the hypotenuse is the diagonal of the square. Applying the Pythagorean Theorem to find the length of the diagonal, we have:

So the length of the diagonal for a square with a side length of is . In general, we could check the length of the diagonal for any square with side length , and we would see that the diagonal length is always .

The diagonal of a rectangle can be thought of as the hypotenuse of a right triangle whose base and height are the length and width of the rectangle, respectively. This means we can use the Pythagorean Theorem to calculate the length of the diagonal for a rectangle:

The area of a rhombus is half the product of the lengths of its diagonals, which here are and . This means

Since both diagonals have whole numbers as their lengths, and , we are looking for to be a whole number that can be divided into 144 to yield a quotient less than . Equivalently,

The quotient is

Since we can multiply both sides by to yield the inequality

,

we know that

,

so we can eliminate 12 and 16.

Also, since , 10 is not correct, as would not be a whole number.

If , then . Both diagonals have lengths that are whole numbers, and is the longer diagonal. 9 is the correct choice.

A rhombus has four sides of equal length. Since Rhombus has a right angle , it follows that, the rhombus being a parallelogram, all four angles are right angles, and, by definition, Rhombus is a square. The length of a diagonal of a square is times the length of a side; since the rhombus has perimeter 80, each side measures one fourth of this, or 20, and diagonal has length .

The referenced rhombus, along with diagonals and , is below.

The four sides of a rhombus have equal measure, so each side has measure one fourth of the perimeter of 48, which is 12.

Since consecutive angles of a rhombus, as with any other parallelogram, are suplementary, and have measure ; the diagonals bisect and into and angles, respectively, to form four 30-60-90 triangles. is one of them; by the 30-60-90 Triangle Theorem, ,

and

.

Since the diagonals of a rhombus bisect each other, .

The sides of a rhombus are all congruent; since the perimeter of Rhombus is 64, each side measures one fourth of this, or 16.

The referenced rhombus, along with diagonal , is below:

Since consecutive angles of a rhombus, as with any other parallelogram, are supplementary, and have measure ; bisects both into angles, making equilangular and, as a consequence, equilateral. Therefore, .

The area of a rhombus is half the product of the lengths of its diagonals, which here are and . This means

Therefore, we need to test each of the choices to find the pair of diagonal lengths for which this holds.

:

Area:

Area:

Area:

Area:

is the correct choice.

The Quadrilateral is shown below with its diagonals and .

. 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. Also, the diagonal that connects the common vertices of the pairs of adjacent sides bisects the other diagonal, making the midpoint of . Therefore,

.

By the 30-60-90 Theorem, since and are the short and long legs of ,

By the 45-45-90 Theorem, since and are the legs of a 45-45-90 Theorem,

.

The diagonal has length

.

This problem can be solved in a variety of ways, however you do it, it may be worth eliminating any options shorter than either side of the quad. The diagonal distance is the hypotenuse of a triangle, so it must be longer than 25 or 60 meters.

Then, we can either find our hypotenuse via Pythagorean Theorem, or our knowledge of Pythagorean Triples.

Using Pythagorean Theorem:

So 65 meters.

Alternatively, recognize the triangle as a 5x/12x/13x triangle

so

Write the formula for finding the area of a square given the diagonal.

Rearrange the equation so that the diagonal term is isolated.

Substitute the known area and simplify.

The diagonal of a square is simply the hypotenuse of a right triangle whose other two sides are the length and width of the square. Because all sides of a square are equal in length, this means the length and width are both , which gives us a right triangle with a base of and a height of , for which the hypotenuse is the diagonal of the square. Applying the Pythagorean Theorem to find the length of the diagonal, we have:

So the length of the diagonal for a square with a side length of is . In general, we could check the length of the diagonal for any square with side length , and we would see that the diagonal length is always .

The diagonal of a rectangle can be thought of as the hypotenuse of a right triangle whose base and height are the length and width of the rectangle, respectively. This means we can use the Pythagorean Theorem to calculate the length of the diagonal for a rectangle:

The area of a rhombus is half the product of the lengths of its diagonals, which here are and . This means

Since both diagonals have whole numbers as their lengths, and , we are looking for to be a whole number that can be divided into 144 to yield a quotient less than . Equivalently,

The quotient is

Since we can multiply both sides by to yield the inequality

,

we know that

,

so we can eliminate 12 and 16.

Also, since , 10 is not correct, as would not be a whole number.

If , then . Both diagonals have lengths that are whole numbers, and is the longer diagonal. 9 is the correct choice.

A rhombus has four sides of equal length. Since Rhombus has a right angle , it follows that, the rhombus being a parallelogram, all four angles are right angles, and, by definition, Rhombus is a square. The length of a diagonal of a square is times the length of a side; since the rhombus has perimeter 80, each side measures one fourth of this, or 20, and diagonal has length .

The referenced rhombus, along with diagonals and , is below.

The four sides of a rhombus have equal measure, so each side has measure one fourth of the perimeter of 48, which is 12.

Since consecutive angles of a rhombus, as with any other parallelogram, are suplementary, and have measure ; the diagonals bisect and into and angles, respectively, to form four 30-60-90 triangles. is one of them; by the 30-60-90 Triangle Theorem, ,

and

.

Since the diagonals of a rhombus bisect each other, .

The sides of a rhombus are all congruent; since the perimeter of Rhombus is 64, each side measures one fourth of this, or 16.

The referenced rhombus, along with diagonal , is below:

Since consecutive angles of a rhombus, as with any other parallelogram, are supplementary, and have measure ; bisects both into angles, making equilangular and, as a consequence, equilateral. Therefore, .

The area of a rhombus is half the product of the lengths of its diagonals, which here are and . This means

Therefore, we need to test each of the choices to find the pair of diagonal lengths for which this holds.

:

Area:

Area:

Area:

Area:

is the correct choice.

The Quadrilateral is shown below with its diagonals and .

. 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. Also, the diagonal that connects the common vertices of the pairs of adjacent sides bisects the other diagonal, making the midpoint of . Therefore,

.

By the 30-60-90 Theorem, since and are the short and long legs of ,

By the 45-45-90 Theorem, since and are the legs of a 45-45-90 Theorem,

.

The diagonal has length

.