point D,E lie on side BC,CA of triangle ABC resp.lines AD,BE,DE bisect angle(BAC,ABC,ADC}resp.Then angle BAC=____radians
To find the measure of angle BAC in radians, we need to use the information given about the bisectors of the angles in triangle ABC.
Let's start by labeling the angles:
- Angle BAC is angle A.
- Angle ABC is angle B.
- Angle BCA is angle C.
According to the information given, lines AD, BE, and DE bisect angles BAC, ABC, and ADC, respectively.
Now, we have two triangles formed: triangle ADE and triangle ABC.
Since DE is a bisector of angle ADC, it divides angle ADC into two equal angles: angle ADE and angle EDC.
Similarly, since AD is a bisector of angle BAC, it divides angle BAC into two equal angles: angle BAD and angle DAC.
Let's label these angles in the respective triangles:
- In triangle ADE, angle DAE = angle EAD = angle ADE = angle EDC = x (let's assume they are all equal to x radians).
- In triangle ABC, angle B = angle BAD + angle DAC = 2x + 2x = 4x.
We know that the sum of the angles in a triangle is always equal to π radians (180 degrees).
So, in triangle ABC, we have:
angle A + angle B + angle C = π radians.
Substituting the values, we get:
x + 4x + x = π radians.
Simplifying:
6x = π radians.
To find the value of x, we divide both sides of the equation by 6:
x = π/6 radians.
Now, we can find the measure of angle BAC (angle A) by substituting the value of x back into the equation:
angle A = x = π/6 radians.
Therefore, the measure of angle BAC is π/6 radians.