Diagonals of a Rhombus: True / false

The diagonals of the rhombus are indeed perpendicular to each other (property of a rhombus)

Therefore, the correct answer is answer A.

The diagonals of the rhombus intersect at their point of intersection, and therefore are not parallel

To solve the problem, let's review a fundamental property of rhombuses:

• In a rhombus, the diagonals have a special property: they intersect each other at right angles (90 degrees) and bisect each other. This means each diagonal cuts the other into two equal halves.

Why is this the case? Consider the fact that a rhombus is a type of parallelogram with all sides of equal length. Therefore, each diagonal acts as a line of symmetry, dividing the rhombus into two congruent triangles. This symmetry ensures that the diagonals not only intersect at right angles but also bisect each other.

In summary, given that the shape in question is a rhombus, we can confidently state that the diagonals do bisect each other.

Therefore, the answer to the problem is Yes.

In a rhombus, all sides are equal, and therefore it is a type of parallelogram. It follows that its diagonals indeed intersect each other (this is one of the properties of a parallelogram).

Therefore, the correct answer is answer A.

First, let's mark the vertices of the rhombus with the letters ABCD, then proceed to draw the diagonals AC and BD, and finally mark their intersection point with the letter E:

Now let's examine the following properties:

a. The rhombus is a type of parallelogram, therefore its diagonals intersect each other, meaning:

$AE=EC=\frac{1}{2}AC\\ BE=ED=\frac{1}{2}BD\\$

b. A property of the rhombus is that its diagonals are perpendicular to each other, meaning:

$AC\perp BD\\ \updownarrow\\ \sphericalangle AEB=\sphericalangle BEC=\sphericalangle CED=\sphericalangle DEA=90\degree$

c. The definition of a rhombus - a quadrilateral where all sides are equal, meaning:

$AB=BC=CD=DA$

Therefore, from the three facts mentioned in: a-c and using the SAS (Side-Angle-Side) congruence theorem, we can conclude that:

d.
$$$\triangle AEB\cong\triangle CEB\cong\triangle AED\cong\triangle CED$(where we made sure to properly and accurately match the triangles according to their vertices in correspondence with the appropriate sides and angles).

Indeed, we found that the diagonals of the rhombus create (together with the rhombus's sides - which are equal to each other) four congruent triangles.

Therefore - the correct answer is answer a.

To solve this problem, let's review the properties of a rhombus:

• A rhombus is a type of quadrilateral with all sides of equal length.
• The diagonals of a rhombus bisect each other at right angles.
• The diagonals of a rhombus do not generally have equal lengths.

However, there is a special case of a rhombus where the diagonals are equal, and this is when the rhombus is a square.

A square is a specific type of rhombus where not only are all sides equal, but also all internal angles are 90 degrees. In such a case, the diagonals are equal, and they also bisect each other at right angles. Therefore, in a square, which is a rhombus, the diagonals are equal.

Therefore, in conclusion, a rhombus can have diagonals that are equal if it is a square.

The correct answer to the problem is: Yes.

To determine if every rhombus is also a square, we begin by examining the definitions of these two shapes:

• A rhombus is defined as a quadrilateral where all four sides have equal length.
• A square is defined as a quadrilateral where all four sides are equal in length and all four angles are right angles (90 degrees).

With these definitions in mind, it is clear that:

• While every square is a rhombus because it has equal side lengths, not all rhombuses have right angles in all four corners.
• A rhombus can have any configuration of angles as long as the opposite angles are equal.
• Therefore, a rhombus does not necessarily satisfy the condition of having four right angles required to be a square.

Consequently, not every rhombus is a square: a square is a special type of rhombus.

Therefore, the solution to the problem is Not true.

To solve this problem, we'll follow these steps:

• Step 1: Identify the properties of a rhombus.
• Step 2: Identify the properties of a square.
• Step 3: Analyze if the properties of a rhombus are met by a square.

Let's work through each step:

Step 1: A rhombus is defined as a quadrilateral where all four sides have equal length.
Step 2: A square is a quadrilateral with all four sides of equal length, and all four angles are 90 degrees.

Step 3: Since both definitions include having all sides of equal length, every square meets this condition of a rhombus. Furthermore, while a square has additional properties like equal angles, it doesn't negate it being a rhombus.

Therefore, every square satisfies the definition of a rhombus.

Concluding, the statement "Every square is a rhombus" is True.