Algebra> The vertices of a triangle are A(-1,7)B(5...

The vertices of a triangle are A(-1,7)B(5,1)C(1,4) the eq of angle bisector

Seema , 5 Years ago

Grade 11

2 Answers

Arun

Last Activity: 5 Years ago

Dear Seema

If the bisector of angle B meets side AC at D then
AD/DC = BA/BC.
BA^2=(5--1)^2+(1--7)^2=36+64=100
BA=10
BC=(5-1)^2+(1-4)^2=16+9=25
BC=5
so
AD/DC=10/5=2.

The co-ordinates of D are
(2/3)C+(1/3)A=(2/3)(1,4)+(1/3)(-1,-7)
=(1/3)(1,1)
=(1/3,1/3).

The slope of DB is m=(1-1/3)/(5-1/3)=(2/3)/(14/3)=1/7.
The line BD passes through B so it has equation
y-1=(1/7)(x-5)
7y-7=x-5
7y=x+2.

ALTERNATIVELY:
The slope of AB is m1=(1--7)/(5--1)=8/6=4/3, the slope of BC is m2=(4-1)/(1-5)= -3/4. m1m2=-1 so AB is perpendicular to BC hence angle B is a right angle. If the slope of BC is m2=tanθ= -3/4 then the slope of the bisector of angle B is
tan(θ+45º)=(tanθ+tan45º)/(1-tanθtan45º)
= (-3/4+1)/(1-(-3/4)*1)
= (1/4)/(7/4)
= 1/7.
The bisector passes through B(5,1) so its equation is
y-1=(1/7)(x-5)
7y=x+2.

Regards

Arun

Rajdeep

Last Activity: 5 Years ago

HELLO THERE!

The process to solve such problems is that, first find the angle between the two lines. The angle bisector will pass through a given point (here, point B). And, when we get the angle between the two lines, we can find out the slope of the angle bisector, as the angle that the angle bisector will make with one of the straight lines will be half of the angle between the two lines. From the slope, and the point through which the angle bisector bisects, we can find out the equation of the angle bisector.

LET’S GO!

We have A(-1, -7)

B(5, 1) and C(1, 4).

Slope of line AB:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{1+7}{5+1} = \frac{4}{3} = m_{1}$

Slope of line CB:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}} = \frac{1-4}{5-1} = -\frac{3}{4} = m_{2}$

Now, to find the angle between the two lines whose slopes are known, we have the formula:

$tan \theta = |\frac{m_{1}-m_{2}}{1+m_{1}m_{2}}| \\\\\implies tan\theta = |\frac{{}\frac{4}{3}+\frac{3}{4}}{1-1}| \\\\\implies tan\theta = undefined \\\\\implies \theta = 90\degree$

Now, when angle between the two straight lines is 90 degrees, the angle between the angle bisector through B and line AB is 45 degrees. But this is the angle that the line makes with the negative X axis. Hence, with the positive X axis, the angle that it makes is 180 – 45 = 135 degrees.

Slope of the angle bisector = tan 135 = -1

Now, from the equation:

$y - y_{1} = m(x - x_{1}) \\\\\implies y - 1 = -1(x - 5) \\\\\implies x + y = -4$

This is the equation of the angle bisector. The angle bisector goes through the point B (5, 1) and its slope is 1, so we found out its equation by the above process. THANKS!