### All Advanced Geometry Resources

## Example Questions

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

Assume quadrilateral is a rhombus. If the perimeter of is and the length of diagonal , what is the length of diagonal ?

**Possible Answers:**

**Correct answer:**

To find the value of diagonal , we must first recognize some important properties of rhombuses. Since the perimeter is of is , and by definition a rhombus has four sides of equal length, each side length of the rhombus is equal to . The diagonals of rhombuses also form four right triangles, with hypotenuses equal to the side length of the rhombus and legs equal to one-half the lengths of the diagonals. We can therefore use the Pythagorean Theorem to solve for one-half of the unknown diagonal:

, where is the rhombus side length, is one-half of the known diagonal, and is one-half of the unknown diagonal. We can therefore solve for :

is therefore equal to . Since represents one-half of the unknown diagonal, we need to multiply by to find the full length of diagonal .

The length of diagonal is therefore

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

Assume quadrilateral is a rhombus. If the area of is square units, and the length of diagonal is units, what is the length of diagonal ?

**Possible Answers:**

**Correct answer:**

This problem relies on the knowledge of the equation for the area of a rhombus, , where is the area, and and are the lengths of the individual diagonals. We can substitute the values that we know into the equation to obtain:

Therefore, our final answer is that the diagonal

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

If the area of a rhombus is , and one of the diagonal lengths is , what is the length of the other diagonal?

**Possible Answers:**

**Correct answer:**

The area of a rhombus is given below.

Substitute the given area and a diagonal. Solve for the other diagonal.

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

If the area of a rhombus is , and a diagonal has a length of , what is the length of the other diagonal?

**Possible Answers:**

**Correct answer:**

The area of a rhombus is given below. Plug in the area and the given diagonal. Solve for the other diagonal.

### Example Question #131 : Plane Geometry

The area of a rhombus is . The length of a diagonal is twice as long as the other diagonal. What is the length of the shorter diagonal?

**Possible Answers:**

**Correct answer:**

Let the shorter diagonal be , and the longer diagonal be . The longer dimension is twice as long as the other diagonal. Write an expression for this.

Write the area of the rhombus.

Since we are solving for the shorter diagonal, it's best to setup the equation in terms , so that we can solve for the shorter diagonal. Plug in the area and expression to solve for .

### Example Question #2 : How To Find The Length Of The Diagonal Of A Rhombus

is a rhombus with side length . Diagonal has a length of . Find the length of diagonal .

**Possible Answers:**

**Correct answer:**

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle to find the length of diagonal . From the problem, we are given that the sides are and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

### Example Question #11 : Finding Sides

is a rhombus. and . Find .

**Possible Answers:**

**Correct answer:**

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle to find the length of diagonal . From the problem, we are given that the sides are and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

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

is a rhombus. and . Find .

**Possible Answers:**

**Correct answer:**

A rhombus is a quadrilateral with four sides of equal length. Rhombuses have diagonals that bisect each other at right angles.

Thus, we can consider the right triangle to find the length of diagonal . From the problem, we are given that the sides are and . Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

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

is a rhombus. , , and . Find .

**Possible Answers:**

**Correct answer:**

Thus, we can consider the right triangle and use the Pythagorean Theorem to solve for . From the problem:

Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

Using the quadratic formula,

With this equation, we get two solutions:

Only the positive solution is valid for this problem.

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

is a rhombus. , , and . Find .

**Possible Answers:**

**Correct answer:**

Thus, we can consider the right triangle and use the Pythagorean Theorem to solve for . From the problem:

Because the diagonals bisect each other, we know:

Using the Pythagorean Theorem,

Factoring,

and

The first solution is nonsensical for this problem.

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