<Question>
What is the number of square units in the area of a triangle whose sides measure 5, 5 and 6 units?
</Question>

<think>
<Plan_1>
Hmm, let's see. I need to figure out the area of this triangle with sides 5, 5, and 6 units. Since two sides are equal, it's an isosceles triangle, right? The two sides of 5 units are the legs, and the base is 6 units. But wait, actually, in an isosceles triangle, the two equal sides are called the legs, and the unequal side is the base.
So here, the base is 6, and the two legs are 5 each.

But how do I find the area? The formula for the area of a triangle is (base * height)/2. I know the base is 6, but I need the height. Since it's an isosceles triangle, if I drop a perpendicular from the apex (the vertex opposite the base) to the base, it will bisect the base into two equal parts. So each half of the base will be 3 units. Then, the height can be found using the Pythagorean theorem in one of the right triangles formed by the height, half the base, and one of the equal sides.
Let me write that down.

So, the height h can be found using the Pythagorean theorem: h² + 3² = 5².

<prompt_1.1> Use the Pythagorean theorem to calculate the height of the triangle as per the above plan. </prompt_1.1>
-----

Alternatively, maybe I can use Heron's formula to verify. Heron's formula says that the area of a triangle with sides a, b, c is sqrt[s(s-a)(s-b)(s-c)], where s is the semi-perimeter. Let's compute that.

<prompt_1.2> Use Heron's formula to calculate the area of the triangle as per the above plan. </prompt_1.2>

Alternatively, since it's an isosceles triangle, another way to calculate the area is (base * height)/2. We already have the base as 6, and height as 4.8, so (6*4.8)/2 = 14.4. So all methods lead to the same answer.
Therefore, the area is 14.4 square units.

Wait, just to confirm once more, maybe using coordinates. Let me place the base of the triangle on the x-axis, from (0,0) to (6,0).
Then the apex is at (3, h). The distance from (3, h) to (0,0) is 5 units. So sqrt[(3-0)^2 + (h -0)^2] = 5. So sqrt(9 + h²) =5.

<prompt_1.3> Use coordinate geometry to verify the area of the triangle as per the above plan. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
Let's compute that. h squared plus 3 squared equals 5 squared. That is h² + 9 = 25. So h² = 25 - 9 = 16. Therefore, h = sqrt(16) = 4. So the height is 4 units.

Now, plug that into the area formula. Area = (base * height)/2 = (6 * 4)/2 = 24/2 = 12. So the area should be 12 square units.
</execution_1.1>

<execution_1.2>
The sides are 5, 5, 6. The semi-perimeter s = (5 + 5 + 6)/2 = (16)/2 = 8. Then the area is sqrt[8(8-5)(8-5)(8-6)] = sqrt[8*3*3*2] = sqrt[8*9*2] = sqrt[144] = 12.
</execution_1.2>

<execution_1.3>
Squaring both sides: 9 + h² =25. So h²=16, h=4. So height is 4. Then the area is (6*4)/2=12.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but let me check if I did everything correctly. Let me verify the steps again.

<prompt_2.1> Verify the steps and calculations used to find the area of the triangle as per the above plan. </prompt_2.1>
-----

Based on execution_1.3:
Therefore, all methods lead to the area being 12 square units. So I think that's the correct answer. I don't see any mistakes in my calculations.
The key was recognizing it's an isosceles triangle, using the Pythagorean theorem to find the height, and then applying the area formula. Heron's formula also confirms the result.

**Final Answer**
The area of the triangle is \boxed{12} square units.
</Plan_2>

<Execution_2>
<execution_2.1>
First, confirming that it's an isosceles triangle with sides 5, 5, 6. Yes, so the two equal sides are 5 each, so the base is 6. Dropping the height splits the base into two segments of 3 each. Then using Pythagoras: 5² = 3² + h². 25 = 9 + h². So h² = 16, h = 4. Then area is (6 * 4)/2 = 12.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, the area of the triangle is $\boxed{12}$ square units.
</Final_answer>
</think>