Triangle Dimensions Calculation

The task is to find the base and height of a triangular flag given their ratio and the flag's area.

1. Define Triangle Ratio and Area

Let the base be $b$ and the height be $h$. Given:

• Ratio: $\frac{b}{h} = \frac{5}{4}$
• Area: $A = 160$ cm²

Using the ratio, we can express the base and height in terms of a common factor, $x$. Let $b = 5x$ and $h = 4x$. This maintains the 5:4 ratio.

2. Apply Triangle Area Formula

The area $A$ of a triangle is given by the formula $A = \frac{1}{2} \times b \times h$. Substitute the expressions for $b$ and $h$, and the given area:

$160 \text{ cm}^2 = \frac{1}{2} \times (5x) \times (4x)$

3. Calculate Common Factor x

Simplify the equation:

$160 = \frac{1}{2} \times 20x^2$

$160 = 10x^2$

Solve for $x^2$:

$x^2 = \frac{160}{10}$

$x^2 = 16$

Solve for $x$. Since length must be positive, we take the positive square root:

$x = \sqrt{16}$

$x = 4$

4. Determine Base and Height Values

Substitute the value $x=4$ back into the expressions for base and height:

• Base: $b = 5x = 5 \times 4 = 20$ cm
• Height: $h = 4x = 4 \times 4 = 16$ cm

5. Verify Calculated Dimensions

Confirm the calculated dimensions meet the problem's conditions:

• Ratio check: The ratio $\frac{b}{h} = \frac{20}{16} = \frac{5}{4}$, which is correct.
• Area check: The area $A = \frac{1}{2} \times 20 \text{ cm} \times 16 \text{ cm} = 10 \times 16 = 160 \text{ cm}^2$, which is also correct.

Thus, the base is 20 cm and the height is 16 cm.