A triangle has corners at points A, B, and C. Side AB has a length of #21 #. The distance between the intersection of point A's angle bisector with side BC and point B is #7 #. If side AC has a length of #14 #, what is the length of side BC?

Answer 1

#BC=11 2/3" units"#

Let D be the point on BC where the angle bisector from A, intersects with BC

Then BC = BD + DC

We know BD = 7 and require to find DC.

Applying the #color(blue)"Angle bisector theorem"# to the triangle.
#color(red)(bar(ul(|color(white)(2/2)color(black)((AB)/(AC)=(BD)/(DC))color(white)(2/2)|)))#

substitute known values into the equation.

#rArr21/14=7/(DC)#
#color(blue)"cross multiply"#
#rArr21DC=7xx14#
#rArrDC=(7xx14)/21=14/3=4 2/3#
#rArrBC=7+4 2/3=11 2/3" units"#
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Answer 2

To find the length of side BC, you can use the Angle Bisector Theorem, which states that in a triangle, the length of the side opposite a given angle is proportional to the lengths of the other two sides.

Let DD be the point where the angle bisector of angle AA intersects side BCBC. According to the given information, ADAD bisects angle AA.

Given that the length of side ABAB is 21 and the length of side ACAC is 14, we can set up the following proportion:

BDDC=ABAC\frac{BD}{DC} = \frac{AB}{AC}

Substituting the given values, we get:

BDDC=2114=32\frac{BD}{DC} = \frac{21}{14} = \frac{3}{2}

Now, let xx be the length of BDBD, so DCDC would be 2x2x since BDBD and DCDC are in the ratio 3:23:2.

Since the distance between the intersection of point AA's angle bisector with side BCBC and point BB is 7, we have:

x+7=21x + 7 = 21

Solving for xx:

x=217=14x = 21 - 7 = 14

Therefore, the length of side BCBC, which is BD+DCBD + DC, is 14+2(14)=14+28=4214 + 2(14) = 14 + 28 = 42.

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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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