One of corners of a rectangular triangle of a straight line, that is makes 90 ⁰. These are simplifies work a little in comparison with a usual triangle as there is a set of the regularities and theorems allowing to express easily some sizes through others. For example, try to find the bisector of a right angle lowered on a hypotenuse.
It is required to you
- - rectangular triangle;
- - known length of legs;
- - known length of a hypotenuse;
- - known corners and one of the parties;
- - the known lengths of parts into which the bisector divides a hypotenuse.
Instruction
1. First of all find a hypotenuse. Let your hypotenuse will be equal to page. The bisector of a right angle divides a hypotenuse into two, most often unequal, parts. Designate one of them for x, and another at the same time will be equal with - x.
2. It is possible to arrive differently: designate two parts for x and at, the condition x + at = will be satisfied at the same time with, it will need to be considered at the solution of the equation.
3. Use the following theorem: the relations of legs and the relation of adjacent pieces into which the bisector of a right angle divides a hypotenuse are equal. That is divide length of legs at each other and equate to the relation x / (with - x). At the same time you watch that in numerator stood adjacent to x a leg. Solve the received equation and find x.
4. Try to arrive in a different way: express legs through a hypotenuse and a corner α. At the same time the adjacent leg will be equal с*cosα, and opposite – с*sinα. The equation in this case will turn out in the following look: х/(with - x) = с*cosα/с*sinα. After simplification x = s*cosα / (sinα+cosα).
5. Having learned length of pieces into which the bisector of a right angle divided a hypotenuse, find length of the hypotenuse by means of the theorem of sine. The corner between a leg and a bisector is known to you - 45 ⁰, two parties of an internal triangle too.
6. Substitute data in the theorem of sine: x / sin45 ⁰=l/sinα. Having simplified expression, you receive l=2xsinα / √ 2. Substitute the found value x: l=2c*cosα*sinα / √ 2 (sinα+cosα) =c*sin2α/2cos (45 ⁰-α). It is also the bisector of a right angle expressed through a hypotenuse.
7. If you were given legs, you have two options: or find hypotenuse length on Pythagorean theorem according to which the sum of squares of legs is equal to a square of a hypotenuse and solve in the way stated above. Or use the following ready formula: l= √ 2*ab / (a+b), where an and b – lengths of legs.