in the triangle ABC , AD ray bisects the exterior angle of the triangle at A and intersects CB ray at D . if AB = 6 cm , AC = 8 cm , and BC = 5 cm , find the length of BD line segment
To find the length of the line segment BD, we can use the concept of Angle Bisector Theorem.
Step 1: Start by drawing the given triangle ABC and extend the side CB to point D.
Step 2: Now, we know that AD is the bisector of the exterior angle at A. This means that BD divides the exterior angle at A into two equal angles.
Step 3: Since AB = 6 cm and AC = 8 cm, we can use the Angle Bisector Theorem to find the length of BD.
According to the Angle Bisector Theorem, the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the two sides opposite the angle.
In this case, we have:
BD/DC = AB/AC
Substituting the given values, we get:
BD/DC = 6/8
Simplifying, we have:
BD/DC = 3/4
Step 4: Since BD and DC are the segments of the same line, their lengths add up to the length of the whole line. Therefore, we can write:
BD + DC = BC
Substituting the given value BC = 5, we have:
BD + DC = 5
Step 5: We can solve these two equations simultaneously to find the values of BD and DC.
From equation (1): BD/DC = 3/4
Multiplying both sides by DC, we get: BD = (3/4) * DC -- (Equation 2)
Substituting equation 2 into equation 3, we have: (3/4) * DC + DC = 5
Multiplying through by 4 to clear the fraction, we get: 3DC + 4DC = 20
Simplifying, we have: 7DC = 20
Dividing both sides by 7, we get: DC = 20/7
Substituting the value of DC into equation 2, we have: BD = (3/4) * (20/7)
Simplifying, we get: BD = 15/7
Therefore, the length of the line segment BD is 15/7 cm.