In △ A B C, let D be a point on BC such that AD...

Deromediqm

Deromediqm

Answered

2022-07-23

In A B C, let D be a point on BC such that AD bisects A. If A D = 6, B D = 4 and D C = 3, then find AB.

Answer & Explanation

edgarovhg

edgarovhg

Expert

2022-07-24Added 12 answers

Step 1
Let E is point of line AD such that A E B = 90 ° and F is point of line AD such that A F C = 90 ° . Then triangles ABE and ACF are similar, triangles DBE and DCF are similar. Using this facts, one can write A F A E = C F B E = C D B D = 3 4
D F D E = C D B D = 3 4
Step 2
Then E F A E = 1 A F A E = 1 4
E F D E = 1 + D F D E = 7 4
A E D E = E F / D E E F / A E = 7
A D D E = A E D E 1 = 6
D E = A D / 6 = 1
A E = 7 D E = 7
B E 2 = B D 2 D E 2 = 16 1 = 15
A B 2 = A E 2 + B E 2 = 49 + 15 = 64
A B = 8
equissupnica7

equissupnica7

Expert

2022-07-25Added 4 answers

Step 1

With angle bisector theorem you can easily find out that A B A C = 4 3 . By this relationship, you can label sides A B = 4 x , A C = 3 x.
Draw perpendicular DK, DL such that points D, L lies on the sides AB, AC respectively.
You can easily find out that A K D D L A (A. A. S.)
By this, you can label K D = D L = h and A K = A L = a
Step 2
By the Pythagorean theorem, you can get the following equations,
h 2 + a 2 = 36 (1)
h 2 + ( 3 x a ) 2 = 9 (2)
h 2 + ( 4 x a ) 2 = 16 (3)
Expanding (2), (3) and substituting (1) as required will lead you to x = 2
As A B = 4 x , A B = 2 4 = 8

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