Understanding the 1:1:2 Triangle and Its Corresponding Sides

When the angles of a triangle are in the ratio of 1:1:2, the angles can be expressed as x, x, and 2x. This rare configuration forms a specific type of triangle that is both fascinating and valuable to understand. Let's delve into how this triangle works and what the ratio of its corresponding sides is.

Defining the Angles of the Triangle

Given the sum of the angles in a triangle is 180^circ, we can set up the equation:

x x 2x 180^circ

Combining like terms, this simplifies to:

4x 180^circ

Solving for x, we find:

x frac{180^circ}{4} 45^circ

Thus, the angles of the triangle are:

45^circ 45^circ 90^circ

This means the triangle is a right isosceles triangle, where the two angles measuring 45^circ are equal. In a right isosceles triangle, the ratio of the lengths of the sides opposite the angles is a key characteristic of this type of triangle.

Determining the Ratio of the Sides

For a right isosceles triangle, let's denote the sides as follows:

The sides opposite the 45^circ angles can be labeled as a (the two equal sides). The side opposite the 90^circ angle (the hypotenuse) is asqrt{2}.

Therefore, the ratio of the corresponding sides is:

a : a : asqrt{2} 1 : 1 : sqrt{2}

Verification Using the Pythagorean Theorem

To verify this, consider a right triangle with angles 45^circ, 45^circ, and 90^circ. If the equal sides are labeled as a, then the hypotenuse, using the Pythagorean theorem, is:

a^2 a^2 b^2

Simplifying this:

2a^2 b^2, b asqrt{2}

This confirms that the ratio of the sides in a 1:1:2 angle triangle is 1 : 1 : sqrt{2}.

Verification Using the Law of Sines

To further confirm the ratio, we can use the Law of Sines:

frac{sin A}{a} frac{sin B}{b}

Substituting the appropriate values, we get:

frac{sin 45^circ}{1} frac{sin 90^circ}{asqrt{2}}

Solving for the hypotenuse:

frac{1/sqrt{2}}{1} frac{1}{asqrt{2}}, asqrt{2} sqrt{2}

This confirms that the ratio of the sides is indeed 1 : 1 : sqrt{2}.

Conclusion

Understanding the 1:1:2 ratio in a triangle involves recognizing that it forms a right isosceles triangle with angles 45^circ, 45^circ, and 90^circ. The ratio of the corresponding sides of this triangle is 1 : 1 : sqrt{2}. This unique configuration has practical applications in geometry and trigonometry, making it a fundamental concept to grasp.