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Geometry 9-12 / Week 08 / Tuesday
2/6
Week 08 · Circles

Tuesday

Angles on the rim
// The turntable and the spotlight
⏱ about 20 min

Tuesday: Angles on the Rim

Wren stands on the chalk circle with a protractor. "Comet, hold the string from the pin to this spoke mark."

Comet holds it. Wren marks two points on the rim and reads the central angle. "Eighty degrees."

"Now I stand over here," he says, walking to the far side of the rim. He sights both marks. "Forty."

"Half," Comet says. "Is that a coincidence? What can we make of it?"

Wren moves to a new spot on the far rim and measures again. "Still forty. Every spot on this side."

Nova projects the circle with the two angles drawn. "Would you like a hint? Both angles cut off the same arc."

"Center angle equals the arc. Rim angle is half the arc," Wren says. "Our designer, test it on the brace too."

Central and inscribed angles

A circle with an 80 degree arc, its central angle, and an inscribed angle on the far rim on the same arc.

A central angle has its vertex at the center. Its measure equals the measure of its arc.

An inscribed angle has its vertex on the circle and two chords as sides. Its measure is half its arc.

So every inscribed angle on the same arc is equal, and each is half the central angle on that arc.

Thursday is Proof Day, when you see why. Today you use it.

ArcCentral angleInscribed angle
60°60°30°
80°80°40°
120°120°60°
180°180°90°

Three more facts to use

  • An angle in a semicircle is a right angle: its arc is 180°, and half of 180° is 90°.
  • A tangent is perpendicular to the radius at the point where it touches the circle.
  • A radius perpendicular to a chord cuts the chord in half. The radius, half the chord and the distance make a right triangle.
A circle of radius 3 meters with a tangent line touching the rim and the dashed radius to the touching point.

A solved problem: the brace

A brace lies across the turntable 1.8 meters from the center. The radius is 3 meters. How long is the brace?

The radius perpendicular to the brace bisects it. Half the brace, the 1.8 meter distance and a 3 meter radius make a right triangle.

Half the brace = square root of 3² - 1.8² = square root of 5.76 = 2.4. The brace is 2 × 2.4 = 4.8 meters.

Why does this work? The two radii to the brace ends are equal, so the perpendicular from the center is a line of symmetry.

INSCRIBED ANGLES
  • Read the question.
  • Tap your answer.
A circle with radius 5 cm showing an arc of 80 degrees, an inscribed angleWren's marks cut off an 80° arc. What is the inscribed angle on the far rim, in degrees?
A circle with radius 5 cm showing an arc of 120 degrees, an inscribed angleTwo spoke marks cut off a 120° arc. How big is an inscribed angle on that arc?
A circle with radius 5 cm showing an arc of 180 degrees, an inscribed angleA chord is a diameter, so its arc is 180°. How big is an inscribed angle on that arc?
CENTRAL ANGLES, CHORDS AND TANGENTS
  • Read the question.
  • Tap your answer.
A circle with radius 5 cm showing an arc of 70 degrees, a central angle, an inscribed angleAn inscribed angle on the rim measures 35°. What is the central angle on the same arc?
A brace lies 1.8 meters from the center of the 3 meter turntable. How long is it, in meters?
A straightedge rests against the rim as a tangent. What angle does it make with the radius to the touching point?
An inscribed angle cuts off a 100° arc. How many degrees is it? Type the number.
An angle in a semicircle measures how many degrees? Type the number.
A radius perpendicular to a chord does what to the chord? Type one word.
StatementTrue or false?
Every inscribed angle on the same arc has the same measure.?
An inscribed angle is twice its arc.?
A tangent is perpendicular to the radius at the touching point.?
A radius perpendicular to a chord bisects the chord.?
40 × 2 = 80?
WHY THIS EXERCISEThese are the four facts the turntable braces and the spotlight rely on all week.
Try it
On your cardboard circle, mark two rim points and the center. Measure the central angle.
Mark a point on the far rim and measure the inscribed angle to the same two points. Is it half?
Draw the turntable with the brace 1.8 meters from the center. Mark the perpendicular radius, half the brace and the right triangle.

Strong work, designer. Tomorrow is Shop Lab: you measure inscribed angles yourself and fit a square corner into a semicircle.

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