In triangle ABC, A=(1,2) B=(9,8) and C= (-4,14). The bisector of <A intersects BC at point D. Find the coordinates of D.

Question

I need help sketching a diagram and finding answer. :)

Answer 1

I think that easiest way to wok this problem is a combination of an often overlooked theorem and some ideas from linear algebra. The theorem is the Angle Bisector Theorem . For the problem at hand it says:

DB / DC = AB / AC Here DB, DC , AB and AC are the lengths of the corresponding line segments.

The distance formula gives AB = 10 and AC = 13. Thus the ratio AB/AC = DB/DC = 10/13

Also useful will be the quantity f , f = DC/(DC + DB) = 1 /( 1 + DB/DC) = 13/23

The interpretation of f is that f is the fraction of the way along the segment from C to B that point D is.

That is to say: travelling from C to B, we will encounter point D after covering about 56.5% of the trip.

The rest of the analysis is linear algebra.

The x coordinate of D = f ( x coordinate of B - x coordinate of C ) + x coordinate of C

The y coordinate of D = f ( y coordinate of B - y coordinate of C ) + y coordinate of C

Plugging in the numbers the point D is (77 / 23 , 244 / 23 )

Answer 2

AB = 10

AC=16.8

BC=25.5

rounded to one decimal

AB/AC = BD/CD

10/16.8 = 9.55/16

Plotting the points, they almost lie on a straight line.

The triangle is very flat, with angle A about 144 degrees

which means BD will

be smaller but very close to the length of AB

D is down 9.55 along the line from B to C

BC has slope(8-2)/9-1) = 3/4

The line BC increases upward by 3 as it goes to the right by 4

To go back down the line by 9.55 is the same as

y decreasing by 4.09 and x decreasing by 5.46

That's the point D (3.46, 4.91

Answer 3

AC = sqrt((1 + 4)² + (2 - 14)²) = sqrt(169) = 13;

AB = sqrt((9 - 1)² + (8 - 2)²) = sqrt(100) =10.

By property of bisector of ∠A wehave: BD/DC = AB/AC;

Let D(x, y). Then vector BD = < x - 9, y - 8> and vector DC = < - 4 - x, 14 - y>;

(x - 9)/(- 4 - x) = 10/13; 13x - 117 = - 40 - 10x; 23x = 77; x = 77/23;

(y - 8)/(14 - y) = 10/13; 13y - 104 = 140 - 10y; 23y = 244, y = 244/23.

Answer: D(77/23, 244/23)