Question

Find MN. LMNP is a rectangle. Angle MOS= 30°. PS= 20cm. S is the midpoint of side MN.

Answer 1

Solution:

∠MOS=30°

∠POS=90°

∠LOP=180°-∠MOS-∠POS→→→→→[Using Linear pair axiom, LM is a line.]

          =180°-30°-90°

           =60°

In Δ POS

Let, PO=x cm, and , OS=y cm

Using Pythagoras theorem

 x²+y²=20²---------(1)

In Rectangle ,LMNP

LP=MN=r cm

→S is the midpoint of side MN.

MS=SN$=\frac{r}{2}$

$sin30^{\circ}=\frac{MS}{OS}\\\\ \frac{1}{2}=\frac{r}{2y}\\\\r=y\\\\sin60^{\circ}=\frac{LP}{PO}\\\\\frac{\sqrt{3}}{2}=\frac{r}{x}\\\\ r=\frac{\sqrt{3}x}{2}\\\\y=\frac{\sqrt{3}x}{2}$

Substituting value of x,and y in equation (1)

$x^2+(\frac{\sqrt{3}x}{2})^2=20^2\\\\x^2+\frac{3x^2}{4}=400\\\\7x^2=1600\\\\x=\frac{40}{\sqrt{7}}\\\\ x=15.13 \\\\ y=\frac{1.732\times 15.13}{2}$

y=13.09640=13 cm(approx)

As, r=y

So, MN=13 cm

Answer 2

Answer:

MN = 13.09 cm

Step-by-step explanation:

in Δ MON

Sin30° = MS/OS

MS = MN/2  ( S is mid point of MN)

Sin30° = 1/2

=> 1/2 = MN/2OS

=> OS = MN

in ΔLOP

∠LOP = 180° - 30° - 90° = 60°

Sin60° = LP/OP

LP = MN   ( Parallel sides of rectangle are equal)

Sin60° = √3/2

=> √3/2 = MN/OP

=> OP = 2MN/√3

OP² + OS²  = 20²

=> (2MN/√3)² + MN²  = 400

=> 4MN²/3  + MN² = 400

=> 4MN² + 3MN² = 1200

=> 7MN² = 1200

=> MN² = 1200/7

=> MN = 13.09