Question

AB DIVIDES ANGLE DAC the ratio 1 :3 and AB is equal to DB. determine the value of x

Answer 1

Hope it helps...............

Answer 2

The value of x is 90°.

Step-by-step explanation:

We are given that AB divides $\angle$DAC in the ratio 1 : 3 and AB is equal to DB.

We have to determine the value of x.

As it is given in the question that AB divides $\angle$DAC in the ratio 1 : 3, that means;

$\dfrac{\angle DAB}{\angle BAC} = \dfrac{1}{3}$        

$\angle BAC = 3 \angle DAB$

Let $\angle$DAB = y, then the value of $\angle$BAC = 3y ------------- [equation 1]

Now, after observing the figure, it is clear that;

$\angle$CAE + $\angle$BAC + $\angle$BAD = 180°     {beacuse of linear pair}

108° + 3y + y = 180°  

4y = 180° - 108°

4y = 72°

y  =  $\frac{72\° }{4}$ = 18°

This means that $\angle$BAD = y = 18° and $\angle$BAC = 3y = $3 \times 18$ = 54°.

Now, as it is given that AB = DB, which means that $\angle$BDA = $\angle$BAD  {because equal sides have equal opposite angles}

Now, in $\triangle$BAD, applying angle sum property of the triangle we get;

$\angle$BDA + $\angle$BAD + $\angle$ABD = 180°

18° + 18° + $\angle$ABD = 180°

$\angle$ABD = 180° - 36° = 144°

Now, it is stated that the sum of interior angles is equal to the exterior angle, that means;

$\angle$BAC + $\angle$BCA = $\angle$ABD

54° + x = 144°

x = 144° - 54° = 90°

Hence, the value of x is 90°.