Best Answer

You can prove this just from knowing that the three angles of a triangle add to 180 degrees.

Label the ends of the arc A and B, the circle's centre O and the "any other point" P.

We want to show that angle APB is half of angle AOB.

(This uses the notation where you write three letters XYZ to mean the angle formed between straight lines XY and YZ).

Consider the triangle formed by O, B and P. The lengths OB and OP are the same (the radius of the circle). This means that the triangle is isosceles and angles OPB and OBP are the same.

Since POB + OPB + OBP = 180 degrees

and OPB = OBP

we have POB + 2 * OPB = 180

With the same working, the triangle formed by O, A and P gives us

POA + 2 * OPA = 180

Subtract these two equations:

(POB - POA) + 2 * (OPB - OPA) = 180 - 180 = 0

Rearrange:

2 * (OPB - OPA) = POA - POB

If you draw a diagram you will see that OPB - OPA = APB

and POA - POB = AOB

so we have 2 * APB = AOB as required.

Q: Angle subtended by an arc is double the angle subtended by it at any other point on the remaining part of the circle?

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360 - 75 = 285

an angle subtended by an arc is double at the center

It is the subtended angle of the arc

It will be the same angle subtended by its circumference.

An angle subtended at the semicircular arc is 90 degrees.

Related questions

360 - 75 = 285

an angle subtended by an arc is double at the center

It is the subtended angle of the arc

Radius: A line from the center of a circle to a point on the circle. Central Angle: The angle subtended at the center of a circle by two given points on the circle.

It will be the same angle subtended by its circumference.

An angle subtended at the semicircular arc is 90 degrees.

It is the consequence of one of the circle theorems and in some books, it is considered a theorem itself. The underlying proposition is that the angle subtended at the circumference of the circle by any arc of a circle is half the angle subtended at the centre. In the case of a semicircle, the arc is the half circle and the angle at the centre is the one that the diameter makes at the centre of the circle ie 180 degrees. So the angle at the circumference is half that ie 90 degrees.

May things, but the probable answer sought here is a diameter of a circle, at the circumference of the circle.

Yes. It follows from one of the circle theorems which states that the angle subtended in a semicircle is a right angle.

A central angle has its vertex at the center of a circle, and two radii form the Arms. Central angle AOC is described as subtended by the chords AC and by the arc AC. An inscribed angle has its vertex on the circle, and two chords form the arms. Inscribed angle ABC is also described as subtended by the chord AC and by the arc AC.

Let us recall the formula for the circumference of a circle. That one is 2pi r. r is the radius of the circle and 2pi is the angle in radian measure subtended by the entire circle at the centre. If this is so, then any arc length 'l' will be equal to the product of the angle in radian measure subtended by the arc at the centre and the radius.So l = theta r. Say theta is the angle subtended by the arc at the centre.Therefrom, r = l / Theta.

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