# GMAT Math : Calculating the length of the diagonal of a quadrilateral

## Example Questions

### Example Question #1 : Calculating The Length Of The Diagonal Of A Quadrilateral

While walking with some friends on campus, you come across an open grassy rectangular area. You shout "Pythagoras!" and run across the rectangular area from one corner to another. If the rectangle measures  meters by  meters, what distance did you cover?

Explanation:

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

### Example Question #2 : Calculating The Length Of The Diagonal Of A Quadrilateral

Given the area of a square is , what is the diagonal?

Explanation:

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.

### Example Question #3 : Calculating The Length Of The Diagonal Of A Quadrilateral

What is the length of the diagonal for a square with a side length of  ?

Explanation:

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  .

### Example Question #4 : Calculating The Length Of The Diagonal Of A Quadrilateral

Calculate the length of the diagonal for a rectangle with a length of    and a width of  .

Explanation:

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:

### Example Question #5 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has area 72.  The lengths of  and  are both whole numbers, and  is the longer diagonal. Which of the following could be the length of ?

None of the other choices gives a correct answer.

Explanation:

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.

### Example Question #6 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has perimeter 80; . What is the length of  ?

Explanation:

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 .

### Example Question #51 : Rhombuses

Rhombus  has area 56.

Which of the following could be true about the values of  and ?

None of the other responses gives a correct answer.

Explanation:

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.

### Example Question #7 : Calculating The Length Of The Diagonal Of A Quadrilateral

Rhombus  has perimeter 64; . What is the length of  ?

Explanation:

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, .

### Example Question #53 : Rhombuses

Rhombus  has perimeter 48; . What is the length of  ?

Explanation:

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, .

### Example Question #2 : Plane Geometry

Given: Quadrilateral  such that   is a right angle, and diagonal  has length 24.

Give the length of diagonal .

None of the other responses is correct.

Explanation:

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

.