A triangle has corners points A, B, and C. Side AB has a length of #9 #. 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 #14 #, what is the length of side BC?

Answer 1

The length of #BC=46/3=15.3#

Let #X# be the point of intersection of the bisector with #BC#
We apply the sine rule to triangle #ABX#
#(AB)/(sin hat(AXB))=(BX)/sin hat(BAX)#
#9/(sin hat(AXB))=6/sin hat(BAX)#.......#(1)#
Then, we apply the sine rule to triangle #ACX#
#(AC)/(sin hat(AXC))=(XC)/sin hat(CAX)#
#(14)/(sin hat(AXC))=(XC)/sin hat(CAX)#.........#(2)#
Combining equations #(1)# and #(2)#
#hat(BAX)=hat(CAX)#

And

#sin hat(AXB)=sin hat(AXC)# as they are supplementary angles
#sin(pi-theta)=sin theta#
#9/6=14/(XC)#
#XC=(14*6)/9=28/3#

Therefore,

#BC=BX+XC=6+28/3=46/3=15.3#
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Answer 2

To find the length of side BC, we can use the Angle Bisector Theorem. According to this theorem, the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides.

Let D be the point where the angle bisector of angle A intersects side BC. According to the problem, AD is the angle bisector and BD is given as 6 units.

We know that the ratio of BD to DC should be the same as the ratio of AB to AC:

[ \frac{BD}{DC} = \frac{AB}{AC} ]

Given that AB = 9 and AC = 14, we can substitute these values into the equation:

[ \frac{6}{DC} = \frac{9}{14} ]

Cross multiplying gives:

[ 6 \times 14 = 9 \times DC ]

[ 84 = 9 \times DC ]

[ DC = \frac{84}{9} ]

[ DC = 9.33 ]

So, the length of side BC is the sum of BD and DC:

[ BC = BD + DC ]

[ BC = 6 + 9.33 ]

[ BC = 15.33 ]

Therefore, the length of side BC is approximately 15.33 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.

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.

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.

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.

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