Question

in a given figure ABC is an equilateral triangle P and S are midpoint of of arc AB and AC prove PQ=QR=RS

Answer 1

Equal Arcs Means Equal Angles

Step-by-step explanation:

Given,  

Arc AX = Arc XB = Arc AY = Arc YC

We know that,

Equal arcs subtend equal angles on the circumferences.

So, $\angle ABC = \angle ACB = 2x$ (Let)

half the arcs subtend half the angles and equal halves subtend equal angles. The angle denoted by x are all equal.

$\angle APQ = \angle PAX + \angle AXP$  

= $x+x = 2x$

$\angle AQP = 2x$

or $\angle APQ = \angle AQP$

AQ = AP

ΔABC is Equilateral Triangle

We know that,

1.) Sides are Equal for Equilateral triangle  

2.) All angles of equilateral triangle are equal

Let one of the angle of ABC be x

• Then, using angle sum property of triangles

$2x + 2x + 2x = 180$

$6x = 180$

$x = \frac {180}{6}$

x = 30 °

$\angle AQR = 2x = 60, \angle ARQ = 2x = 60$

Thus, RS = RA = AQ = PQ = QR

Hence, PQ = QR = RS PROVED !

Answer 2

Answer:

hope it helps...you mate