Rhombus Practice Problems and Solutions for 9th Grade

Let's remember the different definitions of angles:

Corresponding angles are angles located on the same side of the line that intersects the two parallels and are also situated at the same level with respect to the parallel line they are adjacent to.

Therefore, according to this definition, these are the angles marked with the letter A

Alternate angles are angles located on two different sides of the line that intersects two parallels, and which are also not at the same level with respect to the parallel they are adjacent to.

Therefore, according to this definition, these are the angles marked with the letter B

Let's remember that there are two ways to calculate the area of a rhombus:

The first is the side times the height of the side.

The second is diagonal times diagonal divided by 2.

Since we are given both diagonals, we calculate it the second way:

$\frac{7\times4}{2}=\frac{28}{2}=14$

To solve this problem, we need to find the area of a rhombus given the lengths of its two diagonals.

Given Information:

• Main diagonal (longer diagonal): $d_1 = 7$ cm
• Secondary diagonal (shorter diagonal): $d_2 = 2$ cm

Formula for the Area of a Rhombus:
The area of a rhombus can be calculated using its diagonals with the formula: $A = \frac{1}{2} \times d_1 \times d_2$ where $d_1$ and $d_2$ are the lengths of the two diagonals.

Step-by-Step Solution:

Step 1: Identify the diagonal lengths
From the problem, we have:

• $d_1 = 7$ cm (main diagonal)
• $d_2 = 2$ cm (secondary diagonal)

Step 2: Apply the area formula
Substituting the values into the formula: $A = \frac{1}{2} \times 7 \times 2$

Step 3: Calculate the result
$A = \frac{1}{2} \times 14$ $A = 7 \text{ cm}^2$

Verification:
The diagonals of a rhombus bisect each other at right angles. The area formula $\frac{1}{2} \times d_1 \times d_2$ represents half the product of the diagonals, which geometrically divides the rhombus into four right triangles whose combined area equals the total area of the rhombus.

Therefore, the area of the rhombus is 7 cm².

First, let's remember that according to the properties of a rhombus, all sides of a rhombus are equal,

Therefore, if we define the sides of the rhombus with the letters ABCD,

We can argue that:

AB=BC=CD=DA

We use the perimeter formula:

50 = AB+BC+CD+DA

And we can conclude that
 4AB=50

(We can also use any other side, it doesn't matter in this case because they are all equal.)

We divide by four and reveal that:

AB=BC=CD=DA = 12.5

Now let's remember the formula for the area of a rhombus: the height times the side corresponding to the height.

We are given the length of the external height 8,

Now, we can replace in the formula:

8*12.5=100

In a quadrilateral, all sides are equal, and therefore all are equal to 7.

We will also note that in our quadrilateral it is given that the angles are equal to 90 degrees.

We know that a quadrilateral in which all sides are equal and the angles are equal to 90 degrees is actually a square,

and therefore this quadrilateral is also a square, and we can calculate its area accordingly - 

side*side=area

7*7=49

And that's the solution!