True or False? The angle in a semicircle theorem can be used on a diagram that shows a whole circle, not just one showing a semicircle.

True. All the theorem needs is a diameter with a triangle drawn on it, and a diameter can sit inside a whole circle just as easily. The rest of the circle plays no part.

True or False? The radius has to be drawn to the exact point where the tangent touches the circle for the right angle to appear.

True. A radius drawn to any other point on the circumference will meet the tangent at some other angle, or will not reach it at all.

IGCSEMathsCambridge (CIE)CoreFlashcardsGeometryCircle Theorems

Circle Theorems (Cambridge (CIE) IGCSE Maths: Core): Flashcards

Exam code: 0580 & 0980

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FrontAngle in a Semicircle
A triangle is drawn in a circle so that one side is a diameter and the third vertex is on the circumference. What can you say about its angles?

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Cards in this collection (10)

A triangle is drawn in a circle so that one side is a diameter and the third vertex is on the circumference. What can you say about its angles?
The angle at the third vertex, opposite the diameter, is $90 °$ .
This result is called the angle in a semicircle, because only half of the circle is used.
True or False?
The angle in a semicircle is $90 °$ only when the point on the circumference is halfway along the arc.
False.
The angle is $90 °$ for every position of that point, so the vertex can sit anywhere on the arc between the two ends of the diameter.
Why does the angle in a semicircle theorem need a diameter rather than any chord?
Because a diameter passes through the centre, it splits the circle into two semicircles, with the vertex lying on one of them.
A chord that misses the centre cuts off a segment that is not a semicircle, and the angle there is not $90 °$ .
True or False?
The angle in a semicircle theorem can be used on a diagram that shows a whole circle, not just one showing a semicircle.
True.
All the theorem needs is a diameter with a triangle drawn on it, and a diameter can sit inside a whole circle just as easily.
The rest of the circle plays no part.
In a triangle whose side is a diameter, what do the two angles at the ends of the diameter add up to?
They add up to $90 °$ .
The right angle opposite the diameter uses up half of the triangle's $180 °$ , leaving the other two angles to share the rest.
What angle does a tangent make with the radius at its point of contact?
Exactly $90 °$ , so the tangent and that radius are perpendicular.
This holds at every point of the circle, since a tangent can be drawn at any point on the circumference.
True or False?
A tangent meets any chord of a circle at $90 °$ .
False.
The right angle needs a radius, which passes through the centre, so a chord that misses the centre does not meet the tangent at $90 °$ .
A circle has centre $O$ , and $P$ and $Q$ are two points on its circumference. Fill in the two gaps.
Since $O P$ and $O Q$ are both $______$ of the circle, triangle $O P Q$ is $______$ and its base angles are equal.
The completed sentence is:
Since $O P$ and $O Q$ are both radii of the circle, triangle $O P Q$ is isosceles and its base angles are equal.
Any two radii of the same circle are the same length, so this works for every such triangle.
True or False?
The radius has to be drawn to the exact point where the tangent touches the circle for the right angle to appear.
True.
A radius drawn to any other point on the circumference will meet the tangent at some other angle, or will not reach it at all.
A chord and a tangent meet at a point on a circle. How do you find the angle between that chord and the radius?
Subtract the angle between the chord and the tangent from $90 °$ .
The chord lies inside the right angle that the tangent and the radius already make, so it splits that $90 °$ into two parts.

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