6. Geometry
What is the area of a rectangle with a length of 10 cm and a width of 5 cm?

7. Pythagorean Theorem
In a right-angled triangle, if one leg is 3 cm and the other leg is 4 cm, what is the length of the hypotenuse?



Answer :

Sure! Let's go through each question step-by-step.

### 6. Geometry
Question: What is the area of a rectangle with length 10 cm and width 5 cm?

Solution:
To find the area of a rectangle, we use the formula:

[tex]\[ \text{Area} = \text{length} \times \text{width} \][/tex]

Given:
- Length = 10 cm
- Width = 5 cm

Substitute the given values into the formula:

[tex]\[ \text{Area} = 10 \, \text{cm} \times 5 \, \text{cm} \][/tex]

[tex]\[ \text{Area} = 50 \, \text{cm}^2 \][/tex]

So, the area of the rectangle is \( 50 \, \text{cm}^2 \).

### 7. Pythagorean Theorem
Question: In a right-angled triangle, if one leg is 3 cm and the other leg is 4 cm, what is the length of the hypotenuse?

Solution:
To find the length of the hypotenuse in a right-angled triangle, we use the Pythagorean theorem, which states:

[tex]\[ c^2 = a^2 + b^2 \][/tex]

where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two legs.

Given:
- One leg (\( a \)) = 3 cm
- Other leg (\( b \)) = 4 cm

Substitute the given values into the Pythagorean theorem:

[tex]\[ c^2 = (3 \, \text{cm})^2 + (4 \, \text{cm})^2 \][/tex]

[tex]\[ c^2 = 9 \, \text{cm}^2 + 16 \, \text{cm}^2 \][/tex]

[tex]\[ c^2 = 25 \, \text{cm}^2 \][/tex]

To find \( c \), we take the square root of both sides:

[tex]\[ c = \sqrt{25 \, \text{cm}^2} \][/tex]

[tex]\[ c = 5 \, \text{cm} \][/tex]

So, the length of the hypotenuse is [tex]\( 5 \, \text{cm} \)[/tex].