The Hypotenuse of a right triangle is 65 cm and its base is 60 cm . Find the length of perpendicular and the area of the triangle .

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
We have a right triangle with:
- Hypotenuse (c) = 65 cm
- Base (b) = 60 cm

### Step 2: Use the Pythagorean theorem
The Pythagorean theorem states that in a right triangle:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse, \( a \) is the perpendicular (height), and \( b \) is the base.

### Step 3: Substitute the known values into the equation
We need to find the length of the perpendicular (height), denoted as \( a \). Thus, we can rearrange the equation:
\[ a^2 = c^2 - b^2 \]

Substituting the known values:
\[ a^2 = 65^2 - 60^2 \]

### Step 4: Calculate the squares
Now, calculate \( 65^2 \) and \( 60^2 \):
- \( 65^2 = 4225 \)
- \( 60^2 = 3600 \)

Now substitute these values into the equation:
\[ a^2 = 4225 - 3600 \]

### Step 5: Perform the subtraction
Calculate \( 4225 - 3600 \):
\[ a^2 = 625 \]

### Step 6: Find the length of the perpendicular
To find \( a \), take the square root of both sides:
\[ a = \sqrt{625} = 25 \, \text{cm} \]

### Step 7: Calculate the area of the triangle
The area \( A \) of a right triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values we have:
\[ A = \frac{1}{2} \times 60 \times 25 \]

### Step 8: Perform the multiplication
Calculate:
\[ A = \frac{1}{2} \times 1500 = 750 \, \text{cm}^2 \]

### Final Results
- Length of the perpendicular (height) = 25 cm
- Area of the triangle = 750 cm²

---