# Advanced Geometry : How to find the length of the diagonal of a trapezoid

## Example Questions

### Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

What is the length of the diagonals of trapezoid ? Assume the figure is an isoceles trapezoid.

Explanation:

To find the length of the diagonal, we need to use the Pythagorean Theorem. Therefore, we need to sketch the following triangle within trapezoid :

We know that the base of the triangle has length . By subtracting the top of the trapezoid from the bottom of the trapezoid, we get:

Dividing by two, we have the length of each additional side on the bottom of the trapezoid:

Adding these two values together, we get .

The formula for the length of diagonal  uses the Pythagoreon Theorem:

, where  is the point between  and  representing the base of the triangle.

Plugging in our values, we get:

### Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of both diagonals of this quadrilateral.

Explanation:

All of the lengths with one mark have length 5, and all of the side lengths with two marks have length 4. With this knowledge, we can add side lengths together to find that one diagonal is the hypotenuse to this right triangle:

Using Pythagorean Theorem gives:

take the square root of each side

Similarly, the other diagonal can be found with this right triangle:

Once again using Pythagorean Theorem gives an answer of

### Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of the diagonals of this isosceles trapezoid, with .

Explanation:

To find the length of the diagonals, split the top side into 3 sections as shown below:

The two congruent sections plus 8 adds to 14. , so the two congruent sections add to 6. They must each be 3. This means that the top of the right triangle with the diagonal as a hypotenuse must be 11, since .

We can solve for the diagonal, now pictured, using Pythagorean Theorem:

take the square root of both sides

### Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of the diagonal of the isosceles trapezoid given below.

Explanation:

In order to calculate the length of the diagonal, we first must assume that the height is perpendicular to both the top and bottom of the trapezoid.

Knowing this, we can draw in the diagonal as shown below and use the Pythagorean Theorem to solve for the diagonal.

We now take the square root of both sides:

### Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of diagonal  of the trapezoid.

Explanation:

1) The diagonal  can be found from  by using the Pythagorean Theorem.

2) The length of the base of  has to be found because  is the length of the base of .

3) .

4) Using the Pythagorean Theorem on  to find ,

5) Using the Pythagorean Theorem on  to find ,

### Example Question #3 : How To Find The Length Of The Diagonal Of A Trapezoid

Figure NOT drawn to scale.

Refer to the above diagram, which shows Trapezoid  with diagonal . To the nearest whole number, give the length of .

Explanation:

To illustrate how to determine the correct length, draw a perpendicular segment from  to , calling the point of intersection .

divides the trapezoid into Rectangle  and right triangle  .

Opposite sides of a rectangle are congruent, so .

. The two angles of a trapezoid along the same leg - in particular,  and  - are supplementary, so

By the 30-60-90 Triangle Theorem,

Opposite sides of a rectangle are congruent, so , and

is the hypotenuse of right triangle , so by the Pythagorean Theorem, its length can be calculated to be

Set  and :

### Example Question #4 : How To Find The Length Of The Diagonal Of A Trapezoid

Figure NOT drawn to scale.

Refer to the above diagram, which shows Trapezoid  with diagonal . To the nearest whole number, give the length of .

Explanation:

To illustrate how to determine the correct length, draw a perpendicular segment from  to , calling the point of intersection .

divides the trapezoid into Rectangle  and right triangle  .

Opposite sides of a rectangle are congruent, so .

. The two angles of a trapezoid along the same leg - in particular,  and  - are supplementary, so

By the 30-60-90 Triangle Theorem,

Opposite sides of a rectangle are congruent, so , and

is the hypotenuse of right triangle , so by the Pythagorean Theorem, its length can be calculated to be

Set  and :

### Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Figure NOT drawn to scale.

Refer to the above diagram, which shows Trapezoid  with diagonal . To the nearest whole number, give the length of .

Explanation:

To illustrate how to determine the correct length, draw a perpendicular segment from  to , calling the point of intersection .

divides the trapezoid into Rectangle  and right triangle  .

Opposite sides of a rectangle are congruent, so .

. The two angles of a trapezoid along the same leg - in particular,  and  - are supplementary, so

By the 45-45-45 Triangle Theorem,

and

is the hypotenuse of right triangle , so by the Pythagorean Theorem, its length can be calculated to be

Set  and :