The area of a right- angled triangle is two-thirds of the area of a rectangle. The base of the triangle is 80 percent of the breadth of the reactangle. If the perimeter of the rectangle is 200 cm, what is the height of the triangle?

Options

A. 20 cm
B. 30 cm
C. 15 cm
D. Data inadequate

Explanation

To solve the problem step by step, we will use the information given about the right-angled triangle and the rectangle.

### Step 1: Understand the relationship between the areas
The area of the right-angled triangle is given as two-thirds of the area of the rectangle. We can express this mathematically:
\[ \text{Area of Triangle} = \frac{2}{3} \times \text{Area of Rectangle} \]

### Step 2: Write the formulas for the areas
The area of a right-angled triangle can be expressed as:
\[ \text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
The area of a rectangle is given by:
\[ \text{Area of Rectangle} = \text{Length} \times \text{Breadth} \]

### Step 3: Define the base of the triangle
According to the problem, the base of the triangle is 80% of the breadth of the rectangle. If we let \( b \) be the breadth of the rectangle, then:
\[ \text{Base of Triangle} = 0.8b = \frac{4}{5}b \]

### Step 4: Substitute the base into the area formula
Now we can substitute the base into the area formula for the triangle:
\[ \frac{1}{2} \times \frac{4}{5}b \times h = \frac{2}{3} \times (l \times b) \]

### Step 5: Simplify the equation
Cancelling \( b \) from both sides (assuming \( b \neq 0 \)):
\[ \frac{1}{2} \times \frac{4}{5}h = \frac{2}{3}l \]
This simplifies to:
\[ \frac{2}{5}h = \frac{2}{3}l \]

### Step 6: Solve for height \( h \)
To isolate \( h \), we can multiply both sides by \( \frac{5}{2} \):
\[ h = \frac{5}{2} \times \frac{2}{3}l = \frac{5}{3}l \]

### Step 7: Use the perimeter of the rectangle
The perimeter of the rectangle is given as 200 cm:
\[ \text{Perimeter} = 2(l + b) = 200 \]
This simplifies to:
\[ l + b = 100 \]

### Step 8: Express \( b \) in terms of \( l \)
From the equation \( b = 100 - l \).

### Step 9: Substitute \( b \) back into the height equation
Now we can substitute \( b \) into the equation for the base of the triangle:
\[ h = \frac{5}{3}l \]
And since \( b = 100 - l \):
\[ \text{Base of Triangle} = \frac{4}{5}(100 - l) \]

### Step 10: Solve for \( l \) and \( h \)
Now we can substitute \( b \) back into the perimeter equation:
\[ l + (100 - l) = 100 \]
This does not provide new information, but we can use the relationship between \( h \) and \( l \) to find values.

### Conclusion
Since we have two equations but three unknowns (length \( l \), breadth \( b \), and height \( h \)), we cannot uniquely determine the height \( h \) without additional information.

### Final Answer
The data is inadequate to determine the height of the triangle.