1.
We need to use Heron's formula to verify that we get the same area.
Heron's formula is given as:
![A=\sqrt[]{s(s-a)(s-b)(s-c)}](https://img.qammunity.org/2023/formulas/mathematics/college/t3rmhsk71nltcmgtu237bvtgrmrx89v4xv.png)
where:

In this case we have that a=3, b=4 and c=5. Then:
Step 1: Find s

Step 2: Find s-a

Step 3: Find s-b

Step 4: Find s-c

Step 5: Plug the values in Heron's formula
![\begin{gathered} A=\sqrt[]{6\cdot3\cdot2\cdot1} \\ A=\sqrt[]{36} \\ A=6 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/ay7vbyrx53ixd8hp5ypi0ib13sngxfx9rj.png)
Step 6: Verify that it is the same area
We notice that the area using A=1/2bh and Heron's formula is the same.
2.
Triangle 3, 4, 5 is a right triangle. to prove we use the pythagorean theorem:

If we choose a=3, b=4 and c=5 then:

since the pythagorean theorem holds we conclude that the triangle is a right one.
3.
Now we have a triangle with sides 5, 12 and 13 and we need to prove that this is a right triangle. We are going to use the pythagorean theorem to prove it choosing a=5, b=12 and c=13, we have that:

Since the pythagorean theorem holds we conclude that the triangle with sides 5, 12 and 13 is right triangle.
Now we need to find the area with the formula:

in this case we have:

therefore the area is 30 squared units.
Finally we use Heron's formula:
![A=\sqrt[]{s(s-a)(s-b)(s-c)}](https://img.qammunity.org/2023/formulas/mathematics/college/t3rmhsk71nltcmgtu237bvtgrmrx89v4xv.png)
In this case we have that:

then:

Plugging the values in the formula we have:
![\begin{gathered} A=\sqrt[]{15\cdot10\cdot3\cdot2} \\ A=\sqrt[]{900} \\ A=30 \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/gjczaomnkis6wr2bzdsewmw1emfgcznfz2.png)
Therefore the area is 30 squared units and we get the same result.