The triangle area is [tex]\dfrac{1}{4}[/tex] times the square of the hypotenuse.
Pythagorean theorem was not proven by himself. It was Euclid who proven it for the first time.
Fermat’s last theorem is similar to [tex]\sf{a^2+b^2=c^2}[/tex], but any integer satisfy over the power of 2. One of the interesting facts is Fermat didn’t prove it. 358 years since Andrew Wiles came up with the perfect proof.
The converse of the Pythagorean theorem.
Triangle must satisfy [tex]\sf{\overline{AB}^2=\overline{BC}^2+\overline{CA}^2}[/tex] to become a right triangle.
One of the given condition is [tex]\sf{\overline{BC}=\overline{CA}}[/tex]
We get [tex]\sf{\overline{BC}^2=\overline{CA}^2}[/tex]
Another given condition is [tex]\sf{\overline{AB}^2=2\overline{CA}^2}[/tex]
[tex]\sf{\implies \sf{\overline{AB}^2=\overline{CA}^2+\overline{CA}^2}}[/tex]
[tex]\sf{\implies \overline{AB}^2=\overline{BC}^2+\overline{CA}^2}[/tex] (Right-angled at [tex]\sf{\angle{C}}[/tex])
So we have a right triangle according to the converse of the Pythagorean theorem.
Hence proven.
Interesting facts
[tex]\sf{\overline{AB}}[/tex] is √2 times the other sides.
[tex]\sf{\implies\overline{AB}=\sqrt{2} \;\overline{BC}}[/tex]
[tex]\sf{\implies\overline{AB}=\sqrt{2} \;\overline{CA}}[/tex]
The triangle area is [tex]\dfrac{1}{4}[/tex] times the square of the hypotenuse.
Pythagorean theorem was not proven by himself. It was Euclid who proven it for the first time.
Fermat’s last theorem is similar to [tex]\sf{a^2+b^2=c^2}[/tex], but any integer satisfy over the power of 2. One of the interesting facts is Fermat didn’t prove it. 358 years since Andrew Wiles came up with the perfect proof.
Answer:
Given, △ABC is an isosceles triangle,
∴ AC=BC(1)
Also, given that,
(AB)
2
=2(AC)
2
∴ (AB)
2
=(AC)
2
+(AC)
2
(2)
∴ From (1) and (2)
(AB)
2
=(AC)
2
+(BC)
2
Hence, By converse of Pythagoras theorem,
△ABC is an isosceles right-angled triangle. henceproved
hope this helped u
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