Circles - analysis problems

Circles - Analysis Problems

Mathematics 4

August 15, 2011

1 of 21

Circle Analysis

Example 1

A circle with center (2, 1) is tangent to the line y = x+ 2. Find theequation of this circle.

What do we need to solve for? → the radius of the circle.

What do we know?

1. The tangent line is perpendicular to the line passing through theradius and point of tangency.

2. To get the value of the radius, we need to find the coordinates ofthe point of tangency.

2 of 21

Circle Analysis

Example 1

What do we need to solve for?

→ the radius of the circle.

What do we know?

2 of 21

Circle Analysis

Example 1

What do we know?

2 of 21

Circle Analysis

Example 1

What do we know?

2 of 21

Circle Analysis

Example 1

What do we know?

2 of 21

Circle Analysis - Finding the required radius

• Center at (2, 1)

• Tangent to y = x+ 2

3 of 21

• Find the equation of the lineperpendicular to the tangentline and passing through thecenter of the circle.

4 of 21

• Find the intersection of thisline with the original line usingsystems of equations to getthe point of tangency.

5 of 21

• Find the distance from P andC to get the radius.

6 of 21

(x− 2)2 + (y − 1) = 92

• Find the distance from P andC to get the radius.

7 of 21

Recitation Problem

For 2 reci points

Find the standard equation of a circle tangent to y = 2x+ 11 andwhose center is at C(1, 3).

• 1 reci point for the point of tangency

• 1 reci point for the standard equation

8 of 21

Recitation Problem

For 2 reci points

Find the standard equation of a circle tangent to y = 2x+ 11 andwhose center is at C(1, 3).

• 1 reci point for the point of tangency → P (−3, 5)• 1 reci point for the standard equation → (x− 1)2 + (y − 3)2 = 20.

9 of 21

Circle Analysis

Example 2

Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use an algebraic approach.

What do we need to solve for? → the radius and center of the circle.

What do we know?

1. The standard equation of the circle is (x− h)2 + (y − k)2 = r2

2. Three different points satisfying this equation.

What do we need to do? → Find the values for h, k and r2.

10 of 21

Circle Analysis

Example 2

→ the radius and center of the circle.

What do we know?

10 of 21

Circle Analysis

Example 2

What do we know?

10 of 21

Circle Analysis

Example 2

What do we know?

10 of 21

Circle Analysis

Example 2

What do we know?

10 of 21

Circle Analysis

Example 2

What do we know?

What do we need to do?

→ Find the values for h, k and r2.

10 of 21

Circle Analysis

Example 2

What do we know?

10 of 21

Circle Analysis - A(0, 4), B(3, 5) and C(7, 3).

Construct 3 equations using the standard equation and each of thethree points.

1. (0− h)2 + (4− k)2 = r2

2. (3− h)2 + (5− k)2 = r2

3. (7− h)2 + (3− k)2 = r2

Equate the equations since they are all equal to r2.

1 = 2 (0− h)2 + (4− k)2 = (3− h)2 + (5− k)2

2 = 3 (3− h)2 + (5− k)2 = (7− h)2 + (3− k)2

11 of 21

1. (0− h)2 + (4− k)2 = r2

2. (3− h)2 + (5− k)2 = r2

3. (7− h)2 + (3− k)2 = r2

1 = 2 (0− h)2 + (4− k)2 = (3− h)2 + (5− k)2

2 = 3 (3− h)2 + (5− k)2 = (7− h)2 + (3− k)2

11 of 21

1. (0− h)2 + (4− k)2 = r2

2. (3− h)2 + (5− k)2 = r2

3. (7− h)2 + (3− k)2 = r2

1 = 2 (0− h)2 + (4− k)2 = (3− h)2 + (5− k)2

2 = 3 (3− h)2 + (5− k)2 = (7− h)2 + (3− k)2

11 of 21

1. (0− h)2 + (4− k)2 = r2

2. (3− h)2 + (5− k)2 = r2

3. (7− h)2 + (3− k)2 = r2

1 = 2 (0− h)2 + (4− k)2 = (3− h)2 + (5− k)2 → 3h+ k = 9 (A)

2 = 3 (3− h)2 + (5− k)2 = (7− h)2 + (3− k)2 → 2h− k = 6 (B)

12 of 21

Solving Equations A and B simultaneously:

3h+ k = 9

2h− k = 6

We get the center to be (3, 0).

Find the radius by substituting (3, 0) to any of the first threeequations we generated.

(0− 3)2 + (4− 0)2 = r2

9 + 16 = r2

Final standard equation: (x− 3)2 + y2 = 25

13 of 21

3h+ k = 9

2h− k = 6

(0− 3)2 + (4− 0)2 = r2

9 + 16 = r2

13 of 21

3h+ k = 9

2h− k = 6

(0− 3)2 + (4− 0)2 = r2

9 + 16 = r2

13 of 21

3h+ k = 9

2h− k = 6

(0− 3)2 + (4− 0)2 = r2

9 + 16 = r2

13 of 21

3h+ k = 9

2h− k = 6

(0− 3)2 + (4− 0)2 = r2

9 + 16 = r2

13 of 21

3h+ k = 9

2h− k = 6

(0− 3)2 + (4− 0)2 = r2

9 + 16 = r2

Final standard equation: (x− 3)2 + y2 = 2513 of 21

Recitation Problem

Reci Problem 2Find the general equation of the circle containing the pointsA(−5, 0), B(1, 0), and C(−2,−3).

x2 + y2 + 4x− 5 = 0

14 of 21

Recitation Problem

Reci Problem 2Find the general equation of the circle containing the pointsA(−5, 0), B(1, 0), and C(−2,−3).

x2 + y2 + 4x− 5 = 0

14 of 21

Circle Analysis

Example 3

Find the standard equation of the circle containing the pointsA(0, 4), B(3, 5) and C(7, 3). Use a geometric approach.

What do we know? → The perpendicular bisectors of chords intersectat the center.

• Find the equation of the perpendicular bisectors of the midpoints.

• Find the intersection of the perpendicular bisectors of themidpoints, which is the center.

• Find the radius by getting the distance from the center to one ofthe points in the circle.

15 of 21

Circle Analysis

Example 3

→ the radius and center of the circle.

15 of 21

Circle Analysis

Example 3

15 of 21

Circle Analysis

Example 3

What do we know?

→ The perpendicular bisectors of chords intersectat the center.

15 of 21

Circle Analysis

Example 3

15 of 21

Circle Analysis

Example 3

15 of 21

Circle Analysis

Example 3

15 of 21

Circle Analysis

Example 3

15 of 21

Circle Analysis

Example 3

15 of 21

• Circle passes throughA(0, 4), B(3, 5) and C(7, 3)

16 of 21

• Find the midpoints of twochords in this circle.

17 of 21

• Find the equation of theperpendicular bisectors passingthrought the midpoints.

18 of 21

• Find the intersection of theperpendicular bisectors.

19 of 21

(x− 3)2 + y2 = 25

• Find the intersection of theperpendicular bisectors.

• Find the radius and constructthe circle equation.

20 of 21

Recitation Problem

Reci Problem 3Find the standard equation of the circle containing the pointsA(2, 8), B(6, 4) and C(2, 0). Use an geometric approach.

(x− 2)2 + (y − 4)2 = 16

21 of 21

Recitation Problem

Reci Problem 3Find the standard equation of the circle containing the pointsA(2, 8), B(6, 4) and C(2, 0). Use an geometric approach.

(x− 2)2 + (y − 4)2 = 16

21 of 21

Circles - analysis problems

Education

Transcript of Circles - analysis problems

10 MVAAETP481975 Tech · a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and c) ﬁnd arc lengths and areas

Chapter 10: Circles - MacHighwayfrink.machighway.com/~dynamicm/gc-geometry-chap10.pdf · problems involving circumference. ... Chapter 10 Circles 521 ... Guided Practice Use Other

Demidovich Problems in Analysis

Analysis of Kachru's Concentric Circles

Grade 7 & 8 Math Circles November 2, 2011 Dimensional Analysis

Problems Cash Flow Analysis

MATHEMATICS - Paterson Public Schools curriculum guides/1...trapezoids, triangles, half-circles and ... within 20 to solve problems, including word problems involving ... doing”

Stochastic Analysis Problems 2011

Objectives: SWBAT find the circumference of circles SWBAT solve problems involving circumference of circles.

CONFIDENCE CIRCLES FOR CORRESPONDENCE ANALYSIS …

Analitic Analysis of Lines & Circles

CIRCLES, CIRCLES · CIRCLES, CIRCLES, CIRCLES Circles, Circles, Circles is a Private Play about a unique human condition. Not being able to concentrate or focus and always feeling

1 The Seven Bridges of K onigsberg Problem · 2019-12-16 · Graph Theory Problems Berkeley Math Circles 2015 Lecture Notes 1.1 Euler’s Analysis of the Bridge Problem Luckily for

jeamland.cajeamland.ca/~mike/math_11/4.4 Solving Problems... · Chapter 4- Oblique Triangle Trigonometry 44: Solvin Problems Usin Obtuse Trian les Three circles of radius 3, 5, and

Real Analysis - Tutorial Problems

Geometry - Circlesmath.cmu.edu/~cargue/arml/archive/13-14/geometry-12-08-13.pdf · Warm-up Tangent circles Angles inside circles Power of a point Facts Problems Solutions Power of

Problems in Mathematical Analysis

Lesson 6: Unknown Angle Problems with Inscribed Angles in ... · Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles . Student Outcomes Use the . inscribed angle theorem.

10/14/20155.3: Circumference and Area of Circles 5.3: Circumferences and Areas of Circles Expectation: G1.6.1: Solve multistep problems involving circumference.

MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area.