The base and the height of a triangle are in the ratio 4: 5. If the area of the triangle is 40`m^(2)` , find its base and height.

Answer

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

### Step 1: Understand the ratio of base and height
The base and height of the triangle are given in the ratio of 4:5. This means if we let the base be \(4x\), then the height will be \(5x\) for some value of \(x\).

### Step 2: Write the formula for the area of a triangle
The formula for the area \(A\) of a triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the area of the triangle is \(40 \, m^2\), we can substitute the values of base and height into this formula.

### Step 3: Substitute the values into the area formula
Substituting the expressions for base and height into the area formula, we have:
\[ 40 = \frac{1}{2} \times (4x) \times (5x) \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 40 = \frac{1}{2} \times 20x^2 \]
This simplifies to:
\[ 40 = 10x^2 \]

### Step 5: Solve for \(x^2\)
To isolate \(x^2\), divide both sides of the equation by \(10\):
\[ x^2 = \frac{40}{10} \]
\[ x^2 = 4 \]

### Step 6: Solve for \(x\)
Taking the square root of both sides gives:
\[ x = 2 \quad (\text{since } x \text{ cannot be negative}) \]

### Step 7: Find the base and height
Now that we have \(x\), we can find the base and height:
- Base = \(4x = 4 \times 2 = 8 \, m\)
- Height = \(5x = 5 \times 2 = 10 \, m\)

### Conclusion
The base of the triangle is \(8 \, m\) and the height is \(10 \, m\).

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