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A triangle has corners at points A, B, and C. Side AB has a length of #15 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #9 #, what is the length of side BC?

Answer 1

Wyatt Anderson

Length of side BC = 9.6

Let the point where the angle bisector intersects with side BC be D

#"using the "color(blue)"angle bisector theorem"#

#(AB)/(AC)=(BD)/(DC)#

#15 / 9 = 6 / (DC)#

#DC = (6*9) / 15 = 3.6#

#BC = BD+DC= 6+3.6 =9.6#

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Answer 2

Violet Aldrich

Using the Angle Bisector Theorem, the ratio of the lengths of the two segments that the angle bisector divides the opposite side of a triangle is equal to the ratio of the lengths of the other two sides. Let's denote the length of side BC as x.

Applying the Angle Bisector Theorem:

( \frac{AC}{BC} = \frac{AB}{AB + BC} )

Given:

( AC = 9 ) ( AB = 15 ) ( AD = 6 )

Substituting the values:

( \frac{9}{x} = \frac{15}{15 + x} )

Cross multiply:

( 9(15 + x) = 15x )

Expand and solve for x:

( 135 + 9x = 15x ) ( 135 = 6x ) ( x = \frac{135}{6} ) ( x = 22.5 )

So, the length of side BC is 22.5 units.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.