Question

(a) Find x in the below diagram. What are such triangles called?




(b) In a triangle ABC, an altitude is dropped from A to BC at D. $\angle$BAC=40°$\angle$BAD=30°,$\angle$B=(2x+4)°,$\angle$C=(4x-k)°.Find x,k and other angles. [4 MARKS]

Answer 1

(a) Solution: 1 Mark
Type of triangle: 1 Mark
(b) Solution: 1 Mark
Correct answer: 1 Mark


(a) For a triangle, sum of all angles = 180${}^o$
Or, (2x - 15${}^o$) + (x + 20${}^o$) + (x + 15${}^o$) = 180${}^o$
Or, 2x + x + x - 15${}^o$ + 20${}^o$ + 15${}^o$ = 180${}^o$
Or, 4x + 20${}^o$ = 180${}^o$
Or, 4x = 160${}^o$
Or, x = $\frac{160}{4}$ = 40${}^o$

The angles of the triangle will be, (2x - 15${}^o$ = 65${}^o$), (x + 20${}^o$ = 60${}^o$) and (x + 15${}^o$ = 55${}^o$)
Since all angles are less than 90${}^o$, it's an 'acute angled triangle' and also all the angles are of different values therefore all the sides will be of different length, hence it is a 'scalene triangle'.


(b


In triangle ABD
$\Rightarrow 2x+4+90+30=180$ [angle sum property ]
$\Rightarrow 2x=180-124$
$\Rightarrow x=\frac{56}{2}$
$\Rightarrow x=28$
2x+4=2$\times$28+4=60°

In triangle ADC
$\Rightarrow 4x-k+90+10=180$
$\Rightarrow 4x-k=180-100$
$\Rightarrow 4\left(28\right)-k=80$ (Substituting the value of $x=28$)
$\Rightarrow k=4\left(28\right)-80$
$\Rightarrow k=112-80$
$\Rightarrow k=32$
4x-k=4$\times$28 -32=80°

$\therefore$ The angles are ${40}^{\circ },{60}^{\circ }and{80}^{\circ }$ for the given triangle.