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On Time

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Right Time

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

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Estimating Angles

How good are you at estimating angles?

Subtended Angles

Age 11 to 14 Challenge Level:

Arun from the National Public School in Bangalore sent us this detailed solution:

It is clearly seen that the angle at $C$ is twice the angle at $D$. Here we have the fundamental theorem with regard to angles in circles -
" the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the circle "

This can be proved as follows :
Given a circle $(C,r)$ in which arc $AB$ subtends angle ($ACB$) at the centre and angle ($ADB$) at any point $D$ on the circle.

To Prove:
(i) angle $(ACB)$ = $2\times$angle $(ADB)$, when arc $AB$ is a minor arc or a semicircle.
(ii) Reflex angle $(ACB) = 2\times$ angle $(ADB)$, when arc $AB$ is a major arc

subtended angles
Construction: Join $AB$ and $DC$. Produce $DC$ to a point $X$ outside the circle.

Clearly, there are 3 cases.
Case I : arc $AB$ is a minor arc [Fig (i)]
Case II : arc $AB$ is a semicircle [Fig (ii)]
Case III : arc $AB$ is a major arc [Fig (iii)]

We know that when one side of a triangle is produced then the exterior angle so formed is equal to the sum of the interior opposite angles.

Therefore,
angle $(ACX)$ = angle $(CAD)$ + angle $(CDA)$ [consider triangle $CAD$]
angle $(BCX)$ = angle $(CBD)$ + angle $(CDB)$ [consider triangle $CBD$]

But,
angle $(CAD)$ = angle $(CDA)$ [since $CD = CA = r$ ] and ,
angle $(CBD)$ = angle $(CDB)$ [since $CD = CB = r$ ]

Therefore,
angle $(ACX) = 2\times$ angle $(CDA)$ and
angle $(BCX) = 2\times$angle $(CDB)$

In Fig (i) and Fig (ii) ,
angle $(ACX)$ + angle $(BCX)$ = $2 \times$ angle $(CDA)$ + $2\times$ angle $(CDB) \Rightarrow$
angle $(ACB)$ = $2$[angle $(CDA)$ + angle $(CDB)$] $\Rightarrow$
angle $(ACB)$ = $2 \times$angle $(ADB)$

in Fig (iii),
angle $(ACX)$ + angle $(BCX)$ = $2\times$ angle $(CDA)$ + $2\times$ angle $(CDB) \Rightarrow$
reflex angle $(ACB)$ = $2$[angle $(CDA)$ + angle $(CDB)$] $\Rightarrow$
reflex angle $(ACB)$ = $2\times $ angle $(ADB)$

Hence proved.
This theorem has 2 main implications:

(i) The angle in a semicircle is a right angle .
i.e. in fig (2), where arc $AB$ is a semicircle, or in other words $ACB$ is a diameter,
angle $(ADB) = \frac{1}{2} \times$ angle $(ACB) = \frac{1}{2}\times$($180$ degrees) = $90$ degrees = a right angle

This holds true for any position of point $C$ on the semicircle.

(ii) Angles in the same segment of a circle are equal.
This means that if an arc subtends $2$ angles, at $2$ different points on the circle, these angles will be equal.

We can prove this, by proving that each of the $2$ angles is equal to half the angle subtended at the centre.