Respuesta :
1.
We need to use Heron's formula to verify that we get the same area.
Heron's formula is given as:
[tex]A=\sqrt[]{s(s-a)(s-b)(s-c)}[/tex]where:
[tex]s=\frac{a+b+c}{2}[/tex]In this case we have that a=3, b=4 and c=5. Then:
Step 1: Find s
[tex]s=\frac{3+4+5}{2}=6[/tex]Step 2: Find s-a
[tex]s-a=6-3=3[/tex]Step 3: Find s-b
[tex]s-b=6-4=2[/tex]Step 4: Find s-c
[tex]s-c=6-5=1[/tex]Step 5: Plug the values in Heron's formula
[tex]\begin{gathered} A=\sqrt[]{6\cdot3\cdot2\cdot1} \\ A=\sqrt[]{36} \\ A=6 \end{gathered}[/tex]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:
[tex]c^2=a^2+b^2[/tex]If we choose a=3, b=4 and c=5 then:
[tex]\begin{gathered} 5^2=3^2+4^2 \\ 25=9+16 \\ 25=25 \end{gathered}[/tex]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:
[tex]\begin{gathered} 13^2=5^2+12^2 \\ 169=25+144 \\ 169=169 \end{gathered}[/tex]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:
[tex]A=\frac{1}{2}bh[/tex]in this case we have:
[tex]A=\frac{1}{2}(5)(12)=30[/tex]therefore the area is 30 squared units.
Finally we use Heron's formula:
[tex]A=\sqrt[]{s(s-a)(s-b)(s-c)}[/tex]In this case we have that:
[tex]s=\frac{5+12+13}{2}=15[/tex]then:
[tex]\begin{gathered} s-a=15-5=10 \\ s-b=15-12=3 \\ s-c=15-13=2 \end{gathered}[/tex]Plugging the values in the formula we have:
[tex]\begin{gathered} A=\sqrt[]{15\cdot10\cdot3\cdot2} \\ A=\sqrt[]{900} \\ A=30 \end{gathered}[/tex]Therefore the area is 30 squared units and we get the same result.
