What is the relationship between the length of the hypotenuse and the length of the legs in a 45 – 45 – 90 triangle?
The correct answer and explanation is:
In a 45-45-90 triangle, the two legs are of equal length, and the hypotenuse is related to the length of the legs in a specific way. The relationship is that the length of the hypotenuse is √2 times the length of each leg.
This type of triangle is an isosceles right triangle, meaning the two non-hypotenuse sides (the legs) are congruent. In such a triangle, the angles between the legs are both 45 degrees. This symmetry leads to a special geometric ratio.
To derive this relationship, consider the Pythagorean theorem, which states that in any right triangle: a2+b2=c2a^2 + b^2 = c^2
Where:
- aa and bb are the lengths of the legs.
- cc is the length of the hypotenuse.
In a 45-45-90 triangle, both legs have the same length, so we can denote them as xx. The Pythagorean theorem then becomes: x2+x2=c2x^2 + x^2 = c^2
Simplifying: 2×2=c22x^2 = c^2
Taking the square root of both sides: 2×2=c2\sqrt{2x^2} = \sqrt{c^2} c=x2c = x\sqrt{2}
Thus, the length of the hypotenuse cc is equal to the length of a leg xx multiplied by 2\sqrt{2}.
For example, if the length of each leg in a 45-45-90 triangle is 5 units, then the hypotenuse will be: c=52≈7.07 units.c = 5\sqrt{2} \approx 7.07 \, \text{units}.
This relationship is crucial when working with 45-45-90 triangles, as it allows you to easily find one side if you know the other, making it a very efficient and predictable type of right triangle in geometry.