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].
