Prove Triangle ABC is a Right Triangle

  • MHB
  • Thread starter anemone
  • Start date
  • Tags
    Triangle
In summary, to prove that a triangle is a right triangle, you can use the Pythagorean theorem or other methods such as the properties of similar triangles or the ratios of sides in special triangles. The requirements for a triangle to be considered a right triangle include having one 90-degree angle and satisfying the Pythagorean theorem. It is possible to prove a triangle is a right triangle with only the lengths of its sides. It is not possible for a triangle to have two right angles as the sum of the angles in any triangle is always 180 degrees.
  • #1
anemone
Gold Member
MHB
POTW Director
3,883
115
Suppose the lengths of the three sides of $\triangle ABC$ are integers and the inradius of the triangle is 1. Prove that the triangle is a right triangle.
 
Mathematics news on Phys.org
  • #2
Let $a=BC,\,b=CA$ and $c=AB$ be the side lengths, $r$ be the inradius and $s=\dfrac{a+b+c}{2}$.

Since the area of the triangle is $rs$, we get $\sqrt{s(s-a)(s-b)(s-c)}=1\cdot s=s$. Then

$(s-a)(s-b)(s-c)=s=(s-a)+(s-b)+(s-c)$

Now $4(a+b+c)=8s=(2s-2a)(2s-2b)(2s-2c)=(b+c-a)(c+a-b)(a+b-c)$.

In $(\bmod 2)$, each of $b+c-a,\,c+a-b$ and $a+b-c$ are the same. So either they are all odd or all even. Since their product is even, they are all even. Then $a+b+c$ is even and $s$ is an integer.

The positive integers $x=s-a,\,y=s-b$ and $z=s-c$ satisfy $xyz=x+y+z$. Suppose $x\ge y\ge z$. Then $yz\le 3$ for otherwise $xyz>3x\ge x+y+z$. This implies $x=3,\,y=2,\,z=1,\,s=3,\,a=3,\,b=4$ and $c=5$.

Therefore, the triangle is a right triangle.
 

FAQ: Prove Triangle ABC is a Right Triangle

What is a right triangle?

A right triangle is a type of triangle that has one angle measuring 90 degrees. It is also known as a right-angled triangle.

How do you prove that a triangle is a right triangle?

To prove that a triangle ABC is a right triangle, you can use the Pythagorean Theorem, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. If the equation a² + b² = c² holds true, then the triangle is a right triangle.

What other methods can be used to prove that a triangle is a right triangle?

Apart from the Pythagorean Theorem, you can also use the converse of the Pythagorean Theorem, which states that if the square of the length of the longest side is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. Additionally, you can use trigonometric ratios, such as sine, cosine, and tangent, to determine if the triangle is a right triangle.

Can a triangle be a right triangle if it has two equal sides?

Yes, a triangle can still be a right triangle even if it has two equal sides. This type of triangle is called an isosceles right triangle. In this case, the two equal sides are the legs of the triangle, and the remaining side is the hypotenuse.

What is the importance of proving that a triangle is a right triangle?

Proving that a triangle is a right triangle is important in geometry because it allows you to accurately determine the measurements and properties of the triangle. It also helps in solving real-world problems, such as calculating distances and heights using trigonometric functions.

Similar threads

Replies
1
Views
1K
Replies
4
Views
1K
Replies
5
Views
2K
Replies
1
Views
1K
Replies
4
Views
964
Replies
5
Views
2K
Replies
6
Views
1K
Replies
1
Views
842
Back
Top