Question

In ΔABC, m∠CAB = 30°, M is the midpoint of

AB so that AB = 2·MC. Find the angles of the triangle. Find AB if BC = 7 ft. (preferably with a picture)

Answer 1

The angles of the ΔABC are:

`m\angle A=30, m\angle B=60, m\angle C=90`

AB= 14 ft

Explanation:

Given:

A triangle ABC, with
`m\angle A=30`°

AB = 2MC

M is the mid-point of AB.

Let AB =
`2x`

Therefore, AM = MB =
`(AB)/(2)=x`

Also, MC =
`(AB)/(2)=x`

∴ AM = MB = MC =
`x`

Now, consider triangle AMC,

∵ AM = MC

∴
`m\angle MAC = \m\angle MCA = 30`° (
`m\angle A=m\angle MAC`)

Now, exterior angle BMC is given as the sum of opposite interior angles of triangle AMC.

`m\angle BMC=m\angle MAC+m\angle MCA\\m\angle BMC=30+30=60`

Consider triangle BMC,

∵ MB = MC

∴
`m\angle MBC = m\angle MCB = a(Let)`

The sum of all interior angles is equal to 180°.

`m\angle BMC+m\angle MBC+m\angle MCB=180\\60+a+a=180\\2a=180-60\\2a=120\\a=(120)/(2)=60`

Therefore,
`m\angle B =a = 60`°

Also,
`m\angle C=m\angle MCA+m\angle MCB = 30+60=90`°

Therefore, the triangle ABC is a special right angled triangle with measures 30° - 60° - 90°.

For a special right angled triangle 30° - 60° - 90°, the hypotenuse is twice the base.

Here, AB is the hypotenuse and BC is the base. So,

`AB=2BC\\AB=2* 7=14\ ft`

Therefore, AB = 14 ft.