Perimeter of a right angled triangle is 60m and length of hypotenuse of right angled triangle is 25m. If base of the right angled triangle is the smallest side, then find length of smallest side.

Options

A. A)15m
B. B)18m
C. C)21m
D. D)25m

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a right-angled triangle with:
- Perimeter = 60 m
- Hypotenuse = 25 m

We need to find the length of the smallest side, which is the base of the triangle.

### Step 2: Set Up the Equation for the Perimeter
The perimeter of a triangle is the sum of the lengths of its sides. Let:
- \( a \) = length of the smallest side (base)
- \( b \) = length of the other side (height)
- \( c \) = length of the hypotenuse = 25 m

The equation for the perimeter is:
\[ a + b + c = 60 \]

Substituting the value of \( c \):
\[ a + b + 25 = 60 \]

### Step 3: Simplify the Equation
Now, we can simplify the equation to find the relationship between \( a \) and \( b \):
\[ a + b = 60 - 25 \]
\[ a + b = 35 \]

### Step 4: Use the Pythagorean Theorem
Since it is a right-angled triangle, we can use the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
Substituting the value of \( c \):
\[ a^2 + b^2 = 25^2 \]
\[ a^2 + b^2 = 625 \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( a + b = 35 \)
2. \( a^2 + b^2 = 625 \)

From the first equation, we can express \( b \) in terms of \( a \):
\[ b = 35 - a \]

Substituting this into the second equation:
\[ a^2 + (35 - a)^2 = 625 \]

### Step 6: Expand and Simplify
Expanding the equation:
\[ a^2 + (35^2 - 70a + a^2) = 625 \]
\[ 2a^2 - 70a + 1225 = 625 \]
\[ 2a^2 - 70a + 600 = 0 \]

### Step 7: Divide the Equation
To simplify, divide the entire equation by 2:
\[ a^2 - 35a + 300 = 0 \]

### Step 8: Use the Quadratic Formula
Now we can use the quadratic formula to find \( a \):
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -35, c = 300 \):
\[ a = \frac{35 \pm \sqrt{(-35)^2 - 4 \cdot 1 \cdot 300}}{2 \cdot 1} \]
\[ a = \frac{35 \pm \sqrt{1225 - 1200}}{2} \]
\[ a = \frac{35 \pm \sqrt{25}}{2} \]
\[ a = \frac{35 \pm 5}{2} \]

### Step 9: Calculate the Values
Calculating the two possible values for \( a \):
1. \( a = \frac{40}{2} = 20 \)
2. \( a = \frac{30}{2} = 15 \)

### Step 10: Determine the Smallest Side
Since we are given that the base is the smallest side, we take:
- \( a = 15 \) m (smallest side)
- \( b = 20 \) m (other side)

### Final Answer
The length of the smallest side (base) is **15 m**.

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