Week 08 · Circles

Thursday

Proof Day: half the arc

// The turntable and the spotlight

⏱ about 20 min

Thursday: Proof Day, Half the Arc

"Yesterday the readings said half," Wren says, chalking a circle on the floor. "Today we say why."

He marks the center O and a point A on the rim. He draws the diameter from A through O to B.

"Then a third point, C," Comet says, marking it. "So angle BAC is inscribed, and the center sits on side AB."

"That is the easy case," Wren says. "The center on one side. What do you notice if I draw radius OC?"

"OA and OC are both radii. Triangle AOC is isosceles," Comet says. "So its base angles match."

Nova projects the triangle with its two equal angles glowing. "Would you like a hint? Look at angle BOC from outside the triangle."

"An exterior angle," Wren says. "It equals the two far angles added. Our designer, finish the proof."

The theorem

An inscribed angle measures half the arc it cuts off. Today we prove the case where the center lies on one side of the angle.

$A circle with diameter AB, an inscribed angle at A reaching to C, the central angle BOC and the arc BC.$

Given: A, B and C lie on a circle with center O, and O lies on side AB of angle BAC. Call the inscribed angle x.

To prove: angle BAC is half of arc BC.

The proof, read first

Draw radius OC. OA = OC, because both are radii of the same circle.
Triangle AOC is isosceles, so its base angles are equal: angle OCA = angle OAC = x.
Angle BOC is an exterior angle of triangle AOC at O, because A, O and B lie on one line.
An exterior angle equals the sum of the two remote interior angles: angle BOC = x + x = 2x.
Angle BOC is a central angle, so its measure equals arc BC. So arc BC = 2x.
Therefore angle BAC = x = half of arc BC.

Every step uses a fact you already own. Radii are equal (week 1). Base angles of an isosceles triangle match (week 4).

The exterior angle equals the two remote angles (week 4). A central angle equals its arc (Monday).

In the other cases the center is inside or outside the angle. Draw the diameter from A and add or subtract two angles like this one.

Two facts that fall out

Angle in a semicircle: the arc is 180°, so the inscribed angle is 90°. Yesterday's carpenter's square.
Opposite angles of a quadrilateral inscribed in a circle add to 180°. Their two arcs make the whole circle, 360°, and half of 360° is 180°.

A circle around and inside a triangle

The three perpendicular bisectors of a triangle's sides meet at one point, equally far from all three vertices (week 4). A circle centered there passes through all three: the circumscribed circle.

The three angle bisectors meet at one point, equally far from all three sides. A circle centered there touches all three sides: the inscribed circle.

With string and chalk you can draw both on any triangular brace. Try it on Saturday.

THE PROOF, IN ORDER

?Draw radius OC, so OA = OC because both are radii
?Angle BOC is a central angle, so arc BC = 2x
?Angle BOC is an exterior angle of triangle AOC, because A, O and B lie on one line
?Triangle AOC is isosceles, so angle OCA = angle OAC = x
?Angle BOC = x + x = 2x, the sum of the two remote interior angles
?Therefore angle BAC = x = half of arc BC

WHY THIS EXERCISEA proof is a chain. Each link is a fact you already proved, placed where it is needed.

Triangle AOC has two equal sides, OA and OC. What kind of triangle is it? Type one word.

Angle BOC lies outside triangle AOC along the line AB. What kind of angle of the triangle is it? Type one word.

Angle BOC has its vertex at the center, so it is this kind of angle. Type one word.

USE THE THEOREM

Read the question.
Tap your answer.

$A circle with radius 5 cm showing an arc of 140 degrees, an inscribed angle$ An inscribed angle cuts off a 140° arc. How many degrees is it?

A quadrilateral is inscribed in a circle. One angle is 110°. What is the opposite angle, in degrees?

Which lines of a triangle meet at the center of its circumscribed circle?

Statement	True or false?
In the proof, OA = OC because both are radii.	?
Angle BOC equals x, the same as angle BAC.	?
An exterior angle of a triangle equals the sum of the two remote interior angles.	?
The inscribed circle of a triangle passes through its three vertices.	?

WHY THIS EXERCISEChecking each claim against the figure is how you read any proof, including your own.

Draw the proof figure: circle, center O, diameter AB, point C on the rim and radius OC. Mark the two equal angles x.

Shop Book: copy the six steps of the proof and draw the figure beside them.Write which earlier week each step came from.Draw a quadrilateral inside a circle. Measure two opposite angles and add them. What do you get?

Clear reasoning, designer. Tomorrow the spotlight sweeps an arc, and you meet a new way to measure angles.

Grown-up checked my answers

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