Diagonals of a Rhombus Practice Problems and Solutions

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.

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.

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.