considered triangle ABC the median AD and CF intersect at right angle at G.If BC = 3 cm and AB= 4 cm if the length of AC is √k and then find k
Answers
Answer:
5
Step-by-step explanation:
Consider ΔABC,
Let us assume D and F are midpoints of AB and BC respectively.
AB = 2AF => AF = AB/2 = 4/2 = 2 cm.
BC = 2CD => CD = BC/2 = 3/2 = 1.5 cm.
Let G be the intersection point of AD and CF. Note that G is the centroid.
Because AD and CF are medians, we know that:
CG = 2GF and AG= 2GD (∵centroid divides each median in the ratio 2:1)
In order to simplify the notation:
CG=2n, GF = n ; AG = 2m , GD = m and AC = √k.
ΔAGF, ΔAGC and ΔCGD are right angled at G.
Now use Pythagoras theorem:
In ΔAGC,
(2m)² + (2n)² = (√k)² ------------ [1]
In ΔAGF,
(2m)² + (n)² = (2)² --------------- [2]
In ΔCGD,
(2n)² + m² = (1.5)² --------------- [3]
Solve for n by 4[3] - [2]
16n² + 4m² = 9.
n² + 4m² = 4
- - -
-----------------------
15n² = 5 => n² = 1/3
----------------------------
When n² = 1/3 ; m² = 11/12
Using this values solve [1]
4m² + 4n² = k
=> 4.11/12 + 4.1/3 = k
=> 15/3 = k
=> k = 5.
