If A is the center of the circle shown above and AB = BC = CD, what is the value of x ?

A. 15
B. 30
C. 45
D. 60
E. 75


PS21278


_________________

Triangle abc is equilateral, therefore all angles are 60. abc = acb = bac = 60.
Similarly, Triangle adc is equilateral, therefore all angles are 60. adc = acd = dac = 60.

Now, angle bad = bac + dac = 120, since each triangle is a reverse image of the other.

Now triangle bad is isosceles, since ab = ad.
Therefore abd = adb = 30, since bad =120.

Hence, B

you can solve the question easily by knowing two important properties of square and rhombus.

1. diagonals bisect each other.
2. each diagonal forms a 90 degree angle with the other diagonal

Look at the diagram below:Given that AB = BC = CD, also since AB is the radius then AB = AC = AD = radius, so we have that: AB = BC = CD = AC = AD, so basically we have two equilateral triangles ABC and ACD with common base of AC (ABC and ACD are mirror images of each other). Line segment BD cuts the angle ABC in half and since all angles in equilateral triangle equal to 60 degrees then x=60/2=30 degrees.

Answer: B.
_________________

bunuel ,

how can we say that line segment BD cuts the angle ABC in half which property is this , can u expalin me?
thanks

ABC and ACD are two equilateral triangles, which are mirror images of each other if you join vertices B and D, the segment BD must cut AC, the common base, in half. Which makes BD perpendicular bisector of AC --> so BO is a hight, median, and bisector of angle ABC (O being intersection point of BD and AC).

Hope it's clear.
_________________

we are given that AB=BC=CD
AB is the radius of the circle, so AB=AD

This makes ABCD a square. Therefore, BD and AC are the diagonals of the square. Angle A, B, C, D = 90 and hence x=45

Why is this not considered that ABCD is a square?

ABCD is not a square it's a rhombus (the diagonals are not equal).
_________________

Adding teh video solution to teh thread.


_________________

Here AB=BC
Where AB is radius.
From figure AC is also radius.
So AB=BC=AC
So ABC is equilateral triangle.
So each side is of 60
So X=60/2
X=30

Posted from my mobile device

Since all of the radii in a circle have the same length, we know that the 3 red lines below all have the same length.



Since we are told that AB = BC = CD, we can say that all 5 red lines below have the same length



This means the two red triangles must be equilateral triangles, which means all of their angles are 60°



Since all 4 sides of the red quadrilateral below have the same length, we know that the quadrilateral is a rhombus


In a rhombus, the diagonals bisect the angles.
So the blue diagonal above, bisects the 60° angle into two equal angles of 30° each, which means x = 30

Answer: B

Cheers,
Brent
_________________

Do we assume that the lines are touching the circle's circumference? Just based on the image?