CIRCLE 2

(Angles at Circle)

MGMP Matematika

SMPK PENABUR Jakarta

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After learning this topic, students are able to :

1. Explain the relationship of the lengths of two diagonals (as chords) of a cyclic quadrilateral.�2. Explain the relationship of the lengths of two secants that intersect outside a circle.�3. Explain the relationship of the length of a secant and a tangent line that meet outside a circle.

Learning Achievement

(Tujuan Pembelajaran)

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Material of circle 2:

(Cakupan Materi)

• Angle properties of a circle
• Angles of intersecting chords
• Angles of intersecting secants
• Angles on cyclic quadrilateral
• Intersecting Chords, secants and tangent
• Ptolemy Theorem
• Angle n-sided polygon

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PowerPoint Presentation

Students are able to :

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1. Know the relationship of diagonal lines formed by a cyclic quadrilateral, the relationship of two chords that meet outside the circle, and the relationship between a chord line and a tangent line that meet outside the circle
2. Use the properties of relationship between lines and circle on problem solving process

Chord AC and BD

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Secant AC and BC

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Tangent AB

D is point of tangency

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Intersecting Chords Theorem

If the diagonals of the quadrilateral AC and BD intersect at P, then:

AP × PC = BP × PD

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Let’s do the proof :

1. Measure the length of the lines AP, CP, BP, and DP with a ruler
2. Calculate the measured lengths into the formula
3. Does your measurement have the same result as the intersecting chords theorem?

AP × CP = BP × DP

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Study the figure!

If AP = 15, PC = 8 and BP = 10, then find the length of BD!

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Answer :

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Study the figure!

If AP = 15, AC = 23 and BD = 22, then find the length of PD!

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Answer :

AP × PC = BP × PD, suppose BP = a, then :

AP × (AC – AP) = a × (BD – a)

15 × (23 – 15) = a × (22 – a)

120 = 22a – a²

a² – 22a + 120 = 0

(a – 10)(a – 12) = 0

a = 10 or a = 12

If

a = 10 then PD = 12, and

a = 12 then PD = 10

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Intersecting Secants Theorem

If the extension of opposite side are intersecting at point P outside the circle, then:

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AP × DP = BP × CP

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AP × DP = BP × CP

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Let’s do the proof :

1. Measure the length of the lines AD, DP, BC, and CP with a ruler
2. Calculate the lines AP and BP
3. Calculate the measured lengths into the formula
4. Does your measurement have the same result as the intersecting secants theorem?

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Study the figure!

If AP = 14, AD = 6 and CP = 7, then find the length of BC!

Answer :

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Intersecting Secant and Tangent Theorem

If the extension of a chord and a tangent of the circle are intersecting at point P outside the circle, then:

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BP × CP = AP²

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BP × CP = AP²

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Let’s do the proof :

1. Measure the length of the lines BC, CP, and AP with a ruler
2. Calculate the measured lengths into the formula
3. Does your measurement have the same result as the intersecting secant and tangent theorem?

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Study the figure!

If AP = 12 and CP = 8, then find the length of BC!

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BC = BP – CP

BC = 18 – 8

BC = 10

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Intersecting Chords

• Architecture & Carpentry: Used to find the radius of an arch when only the height and width are known.
• Navigation: Ship navigator use these to calculate distances across circular lakes or harbors.
• Physics of Collisions: Helpful in predicting if objects moving at constant speeds across a circular space will collide.

Intersecting Secants

• Global Positioning (GPS): Satellites use secant-based calculations to determine the distance between an orbiting body and specific sites on Earth’s surface.
• Bridge Engineering: Designers use secants to calculate the dimensions of curving bridges and arches.
• Telecommunication: Secant lines help model and illustrate how electronic waves travel during calls or Wi-Fi data transfers.

Intersecting Tangents

• Transportation: A bicycle or car wheel touches the road at a single tangent point at any given moments.
• Astronomy: Used to calculate the angle of elevation of stars and planets from Earth.
• Solar Energy: Engineers use tangent function to find the optimal angle for solar panels to capture maximum sunlight.

Intersecting Chords, Secants, and Tangents Real-Life Examples:

WORKSHEET CIRCLE 2

INTERSECTING CHORDS, SECANTS, AND TANGENTS: SEGMENT LENGTHS

”Take my yoke upon you and learn from me, for I am gentle and humble in heart, and you will find rest for your souls. For my yoke is easy and my burden is light.”

(Matthew 11:29–30)

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AP × PC = BP × PD

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AP × DP = BP × CP

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BP × CP = AP²

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THANK YOU

MGMP Matematika SMP PENABUR Jakarta