Question

Please solve q 87 ?

Answer 1

`AB=6` cm

Explanation:

Let `|ABC|` denote area of `DeltaABC`,
let `|FBD|=2a`,
as `BD=DC, => |FDC|=2a, => |FBC|=2a+2a=4a`,
given `AE:ED=1:2`,
`=> |FAC|:|FDC|=1:2, => |FAC|=a`
`=> |ABC|=|FAC|+|FBC|=a+4a=5a`
`=> AF:AB=|FAC|:|ABC|=1:5`
`=> AB=5*AF=5xx1.2=6` cm

Answer 2

`|AB|=6` cm

Explanation:

There's probably a cool trick I'm missing, but I didn't peek at the other answer. When I'm stuck I find putting it on the Cartesian grid is a way to make progress.

Let's say `B(0,0)=O` at the origin, `C(6c,0)`, and `A(6a,6b)`. Out of experience I put a factor of 6 on to try to avoid fractions; we'll see how that goes.

`D` is the foot of the median, so the midpoint of BC,

`D=1/2(B+C) = D(3c,0)`

`E` is one third the way along AD from `A` to `D.` If it was `2/3` it would be the centroid.

`E = A +1/3(D-A) = 2/3 A + 1/3 D = E(4a+c, 4b)`

`F` is the meet of CE and AB.

`F = C + t (E -C) = B+ u(A-B) = uA quad` because `B=O`

`(6c,0) +t(\ ( 4a+c,4b) -(6c,0) \ ) = u (6a,6b)`

Y coordinate first, it's easier: `4bt = 6 b u`

`2t = 3u`

X: `6c + ( 4a+c-6c ) t = 6 au = 2a (3u) = 2a(2t)=4at`

`6 c = 5 c t`

`t = 6/5 , quad quad u = 2/3 t = 4/5`

`|AF|= |F-A|=|uA-A| = |u-1||A| = (1-u)|AB|`

`|AB| = |AF|/{1-u} = |AF|/{1-4/5}= 5 |AF|`

We're given `|AF|=1.2` cm `=6/5` cm so

`|AB|=6` cm

Answer 3

`6`

Explanation:

Second answer for this one.

The statement is claimed for any triangle, so we only need to work it out for one. Let's forget about the 1.2 for a bit and work this out for one of the usual suspects, say 45/45/90 with legs length six.

We'll put the right angle at the origin and the sides along the axes, `A(6,0), B(0,6), C(0,0)`.

AD is the median to BC so D is the midpoint of BC, `D(0,3).`

E on AD such that `AE=1/3 AD`. We go 2/3 along the way from D toward A for each coordinate, so E=`(4,1)`

CE `C(0,0),E(4,1)` intersects AB `A(6,0),B(0,6)` at F.

CE is `x=4y` and AB is `y=6-x` so `y=6-4y` or `5y=6` or `y=6/5`. `quad x=4y=24/5`

`F(24/5, 6/5)`

`|AF|^2 = (6-24/5)^2+(6/5)^2 = 2(6/5)^2`

`|AB|^2=6^2+6^2=2(6^2)`

`|AB|^2/|AF|^2= 5^2`

So in the original question

`|AB|=5 |AF| = 5(1.2)=6`, choice (1)