Question

in a rectangle ABCD ab = 12 cm and angle BAC = 30 . calculate the lengths of side BC AND DIAGONAL AC​

Answer 1

$\bold{\underline{\underline{Answer:}}}$

Length of the diagonal = 24 cm

Length of side BC = 20 cm(approx.)

$\bold{\underline{\underline{Step\:by\:step\:explanation :}}}$

Given :

• ◾ABCD

1. Length of side AB = 12 cm
2. $\angle{BAC}$ = 30 °

To find :

• Length of side BC
• Length of diagonal AC

Solution :

In a rectangle every angle is 90°

Therefore $\bold{\angle{CBA}}$ = 90 °

Δ CBA is a right angled triangle (1)

Now we have the measure of angle BAC = 30 °

Let Side BC = x cm

•°• We can use the trigonometric ratio :

• $\bold{cos\theta}$ = $\bold{\dfrac{Adjacent\:side}{Hypotenuse}}$

Adjacent side = AB = 12 cm

Hypotenuse = AC = x cm

$\bold{\theta}$ = 30°

Block in the values,

$\rightarrow$$\bold{cos30\degree}$ = $\bold{\dfrac{AB}{AC}}$

$\rightarrow$$\bold{cos30\degree}$ = $\bold{\dfrac{1}{2}}$

$\rightarrow$$\bold{\dfrac{1}{2}}$ = $\bold{\dfrac{12}{x}}$

Cross multiplying,

$\rightarrow$$\bold{x=\:12\times\:2}$

$\rightarrow$$\bold{x=24}$

AC = x = 24 cm

AC = hypotenuse = diagonal of the rectangle = 24 cm

We have the,

• length of the hypotenuse, AC which is 24 cm
• Side AB = 12 cm

Now since we know that the ΔCBA is a right angled triangle (1) we can thereby apply Pythagoras theorem and solve for Side BC.

By Pythagoras theorem,

$\bold{(Hypotenuse)^2\:=\:(Side1) ^2\:+\:(Side2)^2}$

$\rightarrow$$\bold{(AC)^2\:=\:(AB) ^2\:+\:(BC)^2}$

Block in the values,

$\rightarrow$$\bold{(24)^2\:=\:(12) ^2\:+\:(BC)^2}$

$\rightarrow$$\bold{576=144+(BC)^2}$

$\rightarrow$$\bold{576-144=(BC)^2}$

$\rightarrow$$\bold{432=(BC)^2}$

$\rightarrow$$\bold{\sqrt{432}=BC}$

$\rightarrow$$\bold{20.7846096908=BC}$

•°• BC = 20 cm [APPROX.]

•°• Length of side BC = 20 cm