In triangle ABC, AD bisects angle A, BD=4 and CD=5. The perimeter of triangle ABC is 29. Find the lengths of AB and AC.
Take a look at the angle bisector theorem.
AB/AC = BD/DC
To find the lengths of AB and AC in triangle ABC, we can use the Angle Bisector Theorem and the given information about the lengths of BD and CD.
The Angle Bisector Theorem states that in a triangle, if an angle bisector divides the opposite side into two segments, then the ratio of the lengths of those segments is equal to the ratio of the lengths of the other two sides of the triangle.
Let's first find the length of AC:
Using the Angle Bisector Theorem:
AC/AB = CD/BD
Substituting the given values:
AC/AB = 5/4
Cross-multiplying:
4 * AC = 5 * AB
Simplifying:
4AC = 5AB
Now let's use the given information about the perimeter of the triangle:
Perimeter of triangle ABC = AB + BC + AC
Substituting the given values:
29 = AB + BC + AC
Since AD bisects angle A, we can infer that BD and CD divide side BC into two equal segments. Let's denote each of those segments as x:
BC = BD + CD = 4 + 5 = 9
Now, we can rewrite the equation for the perimeter:
29 = AB + 9 + AC
Rearranging and substituting 4AC = 5AB:
AB = 29 - 9 - AC
AB = 20 - AC
Now substitute AB = 20 - AC into the equation 4AC = 5AB:
4AC = 5(20 - AC)
4AC = 100 - 5AC
9AC = 100
AC = 100/9
Now, substitute the value of AC back into the equation AB = 20 - AC:
AB = 20 - (100/9)
Simplifying:
AB = (180/9) - (100/9)
AB = 80/9
So, the lengths of AB and AC are 80/9 and 100/9, respectively.