30) In triangle ABC the interior angle B equals 20° and C 40°. The length of theangle bisector through A is 2. What is the difference of the side lengths of BCand AB?A) 1 B) 1.5 C) 2 D) 4 E) Not possible to answer.
Question
- In triangle ABC the interior angle B equals 20° and C 40°. The length of theangle bisector through A is 2. What is the difference of the side lengths of BCand AB?A) 1 B) 1.5 C) 2 D) 4 E) Not possible to answer.
Solution
To solve this problem, we need to use the Angle Bisector Theorem, which states that the ratio of the lengths of the two segments created by an angle bisector in a triangle is equal to the ratio of the lengths of the other two sides of the triangle.
First, we need to find the measure of angle A. Since the sum of the angles in a triangle is 180°, angle A = 180° - 20° - 40° = 120°.
Next, we use the Angle Bisector Theorem. Let's denote the length of side BC as a and the length of side AB as b. Since the angle bisector divides angle A, we have the ratio a/b = sin(20°)/sin(40°).
We also know that the length of the angle bisector through A is 2. This forms a right triangle with sides BC and AB. Using the law of sines, we can write 2/sin(120°) = a/sin(20°) = b/sin(40°).
From these two equations, we can solve for a and b:
a = 2sin(20°)/sin(120°) and b = 2sin(40°)/sin(120°).
The difference of the side lengths of BC and AB is |a - b| = |2sin(20°)/sin(120°) - 2sin(40°)/sin(120°)| = 2|sin(20°) - sin(40°)|/sin(120°).
Calculating the values, we get approximately 1.5. So, the answer is B) 1.5.
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