Tell me about it!
Suppose that in a triangle
Δ
A
B
C
,
A
B
=
8
,
B
C
=
6
,
A
C
=
7
,
B
D
=
3
, and
A
D
=
5
. Then what is the value of
C
D
=
d
. (round to nearest hundredths)
The answer is 5.09.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
4 solutions
How can we do it by Stewart's theorem? I had a compass an' a ruler in hand while I inserted the answer. :D
Log in to reply
D is on side A B with A D = 5 and B D = 3 .
Using Stewart's Theorem,
( B C ) 2 ( A D ) + ( A C ) 2 ( B D ) = ( A B ) [ ( C D ) 2 + ( A D ) ( B D ) ]
Plugging in values. we get:
( 6 ) 2 ( 5 ) + ( 7 ) 2 ( 3 ) = ( 8 ) ( d 2 + 1 5 )
Solving, we get
d 2 = 2 5 . 8 7 5
d ≈ 5 . 0 9
Log in to reply
exactly well done
You must add that D is on the segment A B . I won't report this yet, but you'll have to add it.
Log in to reply
That can be deduced from the fact that ∣ A D ∣ + ∣ D B ∣ = ∣ A B ∣ .
Log in to reply
well i think there is no need to add this as we can easily find it out by using triangle's inequality as mentioned in my solution.
Just straight away apply stewarts theorem
What's D??????????? Question is wrongR!!!!
I have reported this question.
Log in to reply
I do not see anything wrong with this question. D is a point which you have to determine.
I've now reported it too.
By the Triangle Inequality, A D + B D ≥ A B with equality being achieved when △ A D B is degenerate. Thus, since we have 5 + 3 = 8 , point D must be on A B . Substituting into Stewart's Theorem, we get 3 ⋅ 8 ⋅ 5 + d ⋅ 8 ⋅ d = 7 ⋅ 3 ⋅ 7 + 6 ⋅ 5 ⋅ 6 ⟹ 8 d 2 = 2 0 7 ⟹ d ≈ 5 . 0 9
Same as Mardokay Mosazghi , put differently .
It is clear D is on AB.
Using Cos Rule twice , once for angle B and then for CD, we get,
C
D
=
B
D
2
+
B
C
2
−
2
∗
B
D
∗
B
C
∗
C
o
s
B
=
3
2
+
6
2
−
2
∗
3
∗
6
∗
2
∗
8
∗
6
8
2
+
6
2
−
7
2
=
5
.
0
9
first of all we need to determine where does d lie. for this first take d in the interior of triangle , you will get a triangle ADB which does not follow triangles side inequality also the inequality would not be followed if we take this point d on the either sides AC and BC . so d does not lie in the interior of triangle also not on AC and BC so clearly point d lies on AB. now we can easily find out CD by stewart's theorem or by cosine rule.
cos B = ( a 2 + c 2 − b 2 ) / ( 2 a c ) = ( 3 6 + 6 4 − 4 9 ) / 9 6 = 1 7 / 3 2 and that in triangle DBC d 2 = a 2 + m 2 − 2 a m c o s B = 3 6 + 9 − 3 6 ( 1 7 / 3 2 ) = 2 0 7 / 8 . Thus d = ( 2 0 7 / 8 ) = 5 . 0 9 This can also be done with Stewarts theorm.