Whole Earth Curriculum WEC

Lesson 11 — Circles and Coordinate Geometry

Learners prove and apply circle theorems — the inscribed angle is half the central angle, an angle in a semicircle is right, and a tangent is perpendicular to the radius — then use coordinate geometry to describe circles and figures with numbers. They connect both to the long, global history of measuring circles and places.

D05 P3: Intellectual & Cognitive Awareness D05.S3 55 minutes Draft

Objectives

Essential question

What do the theorems of a circle prove about angles and tangents, and how does coordinate geometry let me describe and analyze figures with numbers?

radiuschordtangentinscribed anglecentral anglecircle equationdistance formulamidpoint

Materials

Standard materials

Low-tech / no-cost

Enriched / lab & device

Works in different contexts

A circle with a central angle and an inscribed angle intercepting the same arc, showing the inscribed angle is half the central angle, a tangent line touching the circle at a right angle to the radius, and beside it a circle on coordinate axes with its equation
A circle with a central angle and an inscribed angle intercepting the same arc, showing the inscribed angle is half the central angle, a tangent line touching the circle at a right angle to the radius, and beside it a circle on coordinate axes with its equation

Lesson 11 — Circles and Coordinate Geometry

Summary

Learners prove and apply circle theorems — the inscribed angle is half the central angle, an angle in a semicircle is right, and a tangent is perpendicular to the radius — then use coordinate geometry to describe circles and figures with numbers. They connect both to the long, global history of measuring circles and places.

Objectives

  • Prove and apply theorems about circles and use coordinate geometry to describe and analyze figures. (D05.S3.10.02)

Connection

A wheel, a water tank, a round field, a dome, a full moon — circles are everywhere, and so are the questions we ask about them: what angle does a corner make when it sits on a circle? Why does a line that just grazes a wheel point straight away from its hub? And how do we describe a circle’s place and size precisely enough to build or map it? The answers are a few tight, provable truths — and a coordinate system that turns “this circle, here, this big” into an equation anyone can check. The same coordinates that describe a circle describe every shape, and they began as a way to describe the world itself.

Materials

  • Circle-and-coordinate chart
  • Graph paper and a compass
  • Math journal

Preparation

  • Copy or draw the circle-and-coordinate chart.
  • Retrieval: from Lesson 10, the Pythagorean theorem and right triangles; from Grade 9, coordinates and proof (D05.S3.09.01). Today we prove circle facts and describe figures with coordinates.
  • Prepare a compass and straightedge demonstration.

Facilitator note

This lesson is written to the learner (“you”). The idea to land: a circle’s theorems are provable truths — an inscribed angle is half the central angle on the same arc, an angle in a semicircle is right, and a tangent is perpendicular to the radius — and coordinate geometry turns “this figure, here, this size” into numbers: the circle (x − h)² + (y − k)² = r², and the distance and midpoint formulas. Teach the proofs explicitly (S-011), then let learners verify by measurement.

The global lens: both tools are shared human inventions. The number π — the ratio of a circle’s circumference to its diameter — was approximated in Egypt (256/81), by Archimedes (between 3 10/71 and 3 1/7), and by the Chinese mathematician Zu Chongzhi (355/113, accurate to six decimal places) (S-291). Coordinates set places in proportion as early as Ptolemy’s latitude/longitude (2nd century) (S-299), and the 12th-century geographer al-Idrisi made a celebrated world map with places located by position (S-298); the modern coordinate system associated with Descartes is one chapter in a longer story (S-240). The intellectual lens: a proof explains why the inscribed angle is always half — not just that it looks so. The critical-thinking lens: test each theorem by measurement, and ask whether the measurement is a proof or an example. The egalitarian lens: these are tools for fair building, mapping, and dividing land — shared knowledge for everyone.

Procedure

  1. Recall (5 min). From Lesson 10, what does the Pythagorean theorem let you find? From Grade 9, how do coordinates describe a point’s place?
  2. Prove circle theorems (20 min). On the chart, watch and then retrace:
    • Inscribed angle: a central angle has its vertex at the center; an inscribed angle has its vertex on the circle. Draw both on the same arc and show (using isosceles triangles formed by radii) that the inscribed angle is half the central angle.
    • Angle in a semicircle: if the arc is a semicircle, the central angle is 180°, so the inscribed angle is 90° — any angle on a semicircle is a right angle.
    • Tangent–radius: a tangent touches the circle at one point; the radius to that point is perpendicular to the tangent (if it were not, the line would cut the circle twice). Test each by drawing and measuring.
  3. Coordinate geometry (15 min). A circle with center (h, k) and radius r has the equation (x − h)² + (y − k)² = r², which is just the distance formula squared. The distance formula and midpoint formula let you find lengths and centers from coordinates. Worked example: the circle with center (2, −1) and radius 3 is (x − 2)² + (y + 1)² = 9.
  4. Guided practice (10 min). With a partner: (a) write the equation of a circle with center (0, 0) and radius 5, and name three points on it; (b) find the distance between (−1, 2) and (3, 5), and the midpoint of the segment joining them.
  5. Close (5 min). Say in one sentence how an inscribed angle relates to its central angle, and what the circle equation really is (the distance formula in disguise).

Differentiation

  • Support: Draw and measure one theorem at a time; name radius/diameter/chord/ tangent before proving.
  • Extension: Given an expanded equation like x² + y² − 4x + 6y = 12, complete the squares to find the center and radius.

Assessment

  • Formative (peer + self): Can the learner state and justify the inscribed-angle, semicircle, and tangent–radius theorems, and write the equation of a circle from its center and radius?
  • Portfolio artifact (unit): The circle-and-coordinate chart with the measured verifications and the worked circle equation, added to the geometry toolkit.

Home connection

Find a circle at home — a plate, a clock, a lid. Trace it, draw an angle with its corner on the edge and both sides through the ends of a diameter, and measure it. Is it a right angle? Explain why to someone.

Resources