### All Intermediate Geometry Resources

## Example Questions

### Example Question #81 : Quadrilaterals

Find the length of a side of a rhombus that has diagonals with lengths of and .

**Possible Answers:**

**Correct answer:**

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

Plug in the lengths of the half diagonals to find the length of the rhombus.

Make sure to round to places after the decimal.

### Example Question #251 : Intermediate Geometry

Find the length of a side of a rhombus that has diagonals with lengths of and .

**Possible Answers:**

**Correct answer:**

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

Plug in the lengths of the half diagonals to find the length of the rhombus.

Make sure to round to places after the decimal.

### Example Question #252 : Intermediate Geometry

Find the length of a side of a rhombus that has diagonals with lengths of and .

**Possible Answers:**

**Correct answer:**

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

Plug in the lengths of the half diagonals to find the length of the rhombus.

Make sure to round to places after the decimal.

### Example Question #31 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with lengths of and .

**Possible Answers:**

**Correct answer:**

First, find the lengths of half of each diagonal.

Plug in the lengths of the half diagonals to find the length of the rhombus.

Make sure to round to places after the decimal.

### Example Question #32 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with lengths of and .

**Possible Answers:**

**Correct answer:**

First, find the lengths of half of each diagonal.

Plug in the lengths of the half diagonals to find the length of the rhombus.

Make sure to round to places after the decimal.

### Example Question #31 : How To Find The Length Of The Side Of A Rhombus

Find the length of a side of a rhombus that has diagonals with side lengths of and .

**Possible Answers:**

**Correct answer:**

First, find the lengths of half of each diagonal.

Plug in the lengths of the half diagonals to find the length of the rhombus.

Make sure to round to places after the decimal.

### Example Question #81 : Quadrilaterals

Given: Parallelogram such that .

True or false: Parallelogram must be a rhombus.

**Possible Answers:**

False

True

**Correct answer:**

False

A rhombus is defined to be a quadrilateral with four congruent sides.

Parallelogram gives the lengths of two of its opposite sides to be congruent, but this is characteristic of all parallelograms. No information is given about the other two sides, so the figure need not be a rhombus.

### Example Question #82 : Quadrilaterals

Given: Parallelogram such that .

True or false: Parallelogram must be a rhombus.

**Possible Answers:**

True

False

**Correct answer:**

True

Opposite sides of a parallelogram are congruent, so

and

All four sides are congruent to one another. It follows by definition that Parallelogram is a rhombus.

### Example Question #34 : How To Find The Length Of The Side Of A Rhombus

Given: Quadrilateral with diagonal ; .

True or false: From the information given, it follows that Quadrilateral is a parallelogram.

**Possible Answers:**

True

False

**Correct answer:**

False

Below are two quadrilaterals marked with drawn.

The quadrilateral on the left has four congruent sides and is by definition a rhombus. The quadrilateral on the right is not a rhombus, since not all four sides are congruent.

In both cases, , , and, by the reflexive property, . By the Side-Side-Side Congruence Postulate, it can be proved that in both diagrams.

Therefore, Quadrilateral need not be a rhombus.

### Example Question #259 : Intermediate Geometry

Given: and .

True or false: It follows from the given information that .

**Possible Answers:**

True

False

**Correct answer:**

False

As we are establishing whether or not , then , , and correspond respectively to , , and .

If we examine the sides and the angle of in the congruence statements - , , and - we see that these are two sides and a *nonincluded* angle. But if we examine the sides and angles of -, , and - we see that these are two sides and an *included* angle.

The only congruence postulate dealing with two sides and an angle is the SAS Congruence Postulate, which requires congruence between two sides and the included angle of both triangles. This theorem does not apply, so we cannot prove the triangles congruent.