Understanding the Triangle Problem

The question asks us to find the height of a specific triangle. We are given two pieces of information:

• The area of the triangle is $144$ cm$^2$.
• The base ($b$) corresponding to this height ($h$) is half the length of the height. This can be written as: $b = \frac{h}{2}$.

Our goal is to use this information and the formula for the area of a triangle to calculate the height.

Area Formula and Substitution

The formula for the area ($A$) of a triangle is:

$$A = \frac{1}{2} \times \text{base} \times \text{height}$$

Using variables, this is:

$$A = \frac{1}{2}bh$$

We know that $A = 144$ cm$^2$ and $b = \frac{h}{2}$. Let's substitute the relationship between the base and height into the area formula:

$$A = \frac{1}{2} \left( \frac{h}{2} \right) h$$

Now, simplify this expression:

$$A = \frac{1}{4} h^2$$

Solving for the Height

Now we can substitute the given area ($A = 144$) into the simplified formula and solve for $h$:

$$144 = \frac{1}{4} h^2$$

To isolate $h^2$, multiply both sides of the equation by 4:

$$144 \times 4 = h^2$$

$$576 = h^2$$

To find the height $h$, take the square root of both sides:

$$h = \sqrt{576}$$

Calculating the square root:

$$h = 24$$

So, the height of the triangle is $24$ cm.

Final Answer Verification

Let's check if this height works with the given conditions:

• Height ($h$) = $24$ cm.
• Base ($b$) = $\frac{h}{2} = \frac{24}{2} = 12$ cm.
• Area ($A$) = $\frac{1}{2}bh = \frac{1}{2} \times 12 \times 24 = 6 \times 24 = 144$ cm$^2$.

The calculated area matches the given area, confirming our height is correct.