A Chord of a Circle is Equal to its Radius.

Question: A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point on the major arc.

1. 45
2. 22.5
3. 15
4. 20
5. 30

“A Chord of a Circle is Equal to its Radius.”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review".  To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

It is given in the question that a chord of a circle is equal to its radius. It is asked to find the angle subtended by this chord at a point on the major arc.

Firstly we should note that
The length of the chord equals the radius of the circle.

If we draw two lines from the center of the circle to either end of the chords, we get two more radii, forming an equilateral triangle with the chord.

Therefore, the angle subtended by the chord (or the minor arc created by the chord) at the center is 60 degrees.

We can remember that all three sides of an equilateral triangle are 60 degrees.
So, the angle subtended by the chord on any point on the major arc is 30 degrees.

The correct answer is option E.

Approach Solution 2:

Let us draw a circle with any radius and center O.
Let AO and BO be the 2 radius of the circle
Let AB be the chord equal to the length of the radius.

Now, we will join them to form a triangle.

Here OA = OB = AB
Hence ∆ABO becomes an equilateral triangle.
Draw 2 points C and D on the circle such that they lie on the major arc and minor arc, respectively.
Since ∆ABO is an equilateral triangle, we get ∠AOB = 60°.
For the arc AB, ∠AOB = 2∠ACB as we know that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
∠ACB = 1/2 ∠AOB = 1/2 × 60 = 30°

Hence, E is the correct answer.

Approach Solution 3:

Chord of the Circle = Radius of the Circle
So AOB is an equilateral triangle where, OA=OB=AB
∠AOB = 60°
Now, if 2 angles are made on the same chord, angle at centre is twice the angle at circumference.
∠AOB = 2*∠ACB
60° = 2*∠ACB
∠ACB = 30°

Hence, E is the correct answer.

Suggested GMAT Quant Questions:

• How many factors does 36^2 have?
• A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively
• Is Square Root = Always Positive?
• Running at the Same Constant Rate, 6 Identical Machines can
• If mn ≠ 0 and 25 Percent of n Equals 37 1/2 Percent of m, What is the Value of 12n/m ?
• What is the largest five-digit number which is divisible by 17?
• If x is not equal to zero, and x + 1/x = 3, then what is the value of x^4 + 1/x_4?
• Two pipes together can fill a tank in 35 minutes. The larger pipe alone can fill the tank in 24 minutes less time than the smaller pipe.
• The light in a restroom operates with a 15-minute timer that is reset every time the door opens as a person goes in or out of the room
• In a shop, the marked price of an article is worked out in such a way that it generates a profit of 331/3
• If r and s are the roots of the equation x^2 + bx + c = 0, where b and c are constants, is rs < 0 ?
• The product of two numbers is 2028 and their HCF is 13. What are the number of such pairs?
• Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE?
• How many 3-digit positive integers with distinct digits are there, which are not multiples of 10?
• If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
• What is the smallest five digit number that is divisible by 16, 24, 36 and 54?
• If the population of a town is p in the beginning of any year then it becomes 3+2p in the beginning of the next year
• The following tables show the revenues and costs, in thousands of dollars, for a small company in the year 2007
• In the xy-plane, the vertices of a triangle have coordinates (0,0),(3,3), and (7,0). What is the perimeter of the triangle?
• What is the probability that a student randomly selected from a class of 60 students will be a male who has brown hair?