Name: Chords of a Circle
Uploaded: 2019-07-19
Duration: 2 min 11 s
Description: A secant line to a circle is a line that crosses exactly two points on the circle, while a chord is a line that has its two endpoints on the circle. Watch th...

Question

The numbers $1, 4, 8$ on a clock’s face are joined to make a triangle, in the figure of a clock given below, How many equilateral triangles can we make by joining numbers on the clock’s face?

Accepted Answer

We need to find the angles of the triangle made in the given clock.
Figure is given below as,

We know that a clock is a circle and, a circle is made of $360 °$
The clock has $12$ numbers & each number represents an angle.
Separation between each angle $=$ $\frac{360 °}{12} = 30 °$
Therefore from the given figure ,we can see,
$∠ COB = 30 ° \times 4 = 120 °$
We know the theorem,
The angle subtended by an arc at the Centre is double the angle subtended by it,at any point on remaining part of the circle.
So using this theorem,
$∠ CAB = \frac{1}{2} ∠ COB = \frac{1}{2} \times 120 ° = 60 °$
Now,
$∠ AOB = 30 ° \times 3 = 90 °$
Using the same theorem stated above,
$∠ ACB = \frac{1}{2} \times ∠ AOB = \frac{1}{2} \times 90 ° = 45 °$

Now,in the $△ ABC$
$∠ A + ∠ B + ∠ C = 180 °$ (angle sum property)
$60 ° + ∠ B + 45 ° = 180 ° (∵ ∠ A = 60 °, ∠ C = 45 °)$
$∠ B = 180 ° - (60 ° + 45 °) = 180 ° - 105 ° = 75 °$
Hence all the angles of the triangle are $60 °, 45 °, 75 °$ .
We can make $4$ equilateral triangles by joining the numbers on the clock $(1, 5, 9)$ , $(2, 6, 10)$ , $(3, 7, 11)$ , $(4, 8, 12)$ .

The numbers $1, 4, 8$ on a clock’s face are joined to make a triangle, in the figure of a clock given below, How many equilateral triangles can we make by joining numbers on the clock’s face?

Important Questions on Circles

Learn and Prepare for any exam you want !

The numbers 1,4,8 on a clock’s face are joined to make a triangle, in the figure of a clock given below, How many equilateral triangles can we make by joining numbers on the clock’s face?

Important Questions on Circles

Learn and Prepare for any exam you want !

The numbers $1, 4, 8$ on a clock’s face are joined to make a triangle, in the figure of a clock given below, How many equilateral triangles can we make by joining numbers on the clock’s face?