MODERN GEOMETRYSections 7.1, 7.2

Thompson

ANNOUNCEMENTS

Exam IV (Chapter 7)

Thursday 4/16

Teaching Presentations

Tuesday 4/21 & Thursday 4/23

TEACHING PROJECT

Hippocrates’ Quadrature of the Lune (440 B.C.)

Euclid’s Proof of the Pythagorean Theorem (300

B.C.)

Archimedes Determination of Circular Area (225

B.C.)

Heron’s Formula for Triangular Area (75 A.D)

A Gem from Isaac Newton (1665)

ANGLE / ARC NOTATION

CIRCLE BASICS

CIRCLE BASICS

CIRCLE BASICS

ARC CONGRUENCE

CIRCLE BASICS

Needs three points to identify

PROBLEM SOLVING

The angle addition postulate still holds for arc measure𝑚 𝑅𝑆 +𝑚 𝑆𝑇 =180°

𝑚 𝑅𝑆 = 𝑚∠𝑅𝑃𝑆 =65°

CHALLENGE

1. Construct a segment 𝐴𝐵 of length 2 in.

2. Construct a circular arc 𝐴𝐵 with length of 2in.

A B

A B

ARC LENGTH

x°

A

B

Arc length is only a “portion” or “fraction” of the entire circle

l 𝐴𝐵 = 𝑥

3602𝜋𝑟

PROBLEM SOLVING

50°

A

B

l

AREA OF A SECTOR

AREA OF A SECTOR

x°

A

B

Sector area is only a “portion” or “fraction” of the entire circle

Areasector=𝑥

360𝜋𝑟2

CHALLENGE

Find the exact

center of this

circle!

BISECTING CHORDS

INTERSECTING CHORDS CONJECTURE1.Construct Circle

P2.Construct chord

AB3.Construct an

intersecting chord CD

4.Mark intersection as pt E

5.Measure each of the four segment lengths

A

BC

D

E

INTERSECTING CHORDS

PROBLEM SOLVING

THE “HOURGLASS” COROLLARY

A

BC

D

E

What can be said about ΔACE and ΔBDE?

INSCRIBED ANGLES

INSCRIBED ANGLES CONJECTURE1.Construct Circle P2.Construct inscribed angle ∠RST3.Construct central angle ∠RPT4.Find m∠RST5.Find m∠RPT6.Conjecture?

INSCRIBED ANGLES

THE “STAR TREK” LEMMA

PROOF! Conjecture: m∠RST= 2 m∠RPT

INSCRIBED SEMICIRCLE COROLLARY

PROBLEM SOLVING

PROBLEM SOLVING

CIRCLE SEGMENT

CIRCLE SEGMENTFind the area of the segment

CIRCLE SEGMENT

1. Construct Circle P with radius 2in.

2. Construct central angle of 80°3. Find the area of the segment

produced.

CLASSWORK / HOMEWORK

Page 369

1 – 4, 5 – 30 odd, 39, 43, 48, 57

Page 378

1 – 11 odd, 19 – 22