O O O O T Madas T Madas

A kite ABCD is inscribed in a circle, centre at point O. RBAD =

A, B, C and D are points on a circle whose centre is at

P, Q, R and S are points on the circumference of a circle, centre

A regular enneagon (9 -sided polygon) ABCDEFGHI is shown below. Calculate RICE. I A

Slides: 17

Download presentation

O O O O © T Madas

A kite ABCD is inscribed in a circle, centre at point O. RBAD = 46°. 1. Calculate the size of the other three angles. 2. Explain why the line AC has to be a diameter of the circle. A circle theorem states: C Opposite angles in a cyclic quadrilateral are supplementary (i. e. they add up to 180°) 134° B D ∴ RBCD = 134° RABC and RADC are also supplementary and from the properties of a kite they are also equal O 46° A ∴ RABC = RADC = 90° “An inscribed angle which corresponds to a diameter of a circle is always a right angle” © T Madas

A, B, C and D are points on a circle whose centre is at point O. AB is a diameter, ECF is a tangent to the circle with point C the contact point, AB is parallel to CD, BC = BE and RABD = 20°. Calculate, giving reasons for your answers, the size of: D A 20 ° F 1. RBEC 2. RACF RABD = RBDC = 20° C O 20 ° 20° B E [alternating angles] RBCE = RBDC = 20° [by the alternating segment theorem] RBEC = RBCE = 20° [isosceles triangle] © T Madas

A, B, C and D are points on a circle whose centre is at point O. AB is a diameter, ECF is a tangent to the circle with point C the contact point, AB is parallel to CD, BC = BE and RABD = 20°. D A 20 ° F ° 0 6 O 20 ° 20° B 1. RBEC 2. RACF = RCBA C E Calculate, giving reasons for your answers, the size of: 40° [alternating segment theorem] REBC = 140° [angles in a triangle] RCBD = 40° [angles in a straight line] 140° © T Madas

P, Q, R and S are points on the circumference of a circle, centre at point O, with PR being a diameter. ROSQ = x and RQPR = 3 x [all angles measured in degrees] Express in terms of x, where appropriate, the following angles: 1. ROSR 2. RORS 3. RSOR 4. RSQR 5. RPRQ 6. RPRQ Q P 3 x O R x 3 x S Inscribed angles which correspond to the same arc are equal. © T Madas

P, Q, R and S are points on the circumference of a circle, centre at point O, with PR being a diameter. ROSQ = x and RQPR = 3 x [all angles measured in degrees] Express in terms of x, where appropriate, the following angles: 1. ROSR 2. RORS 3. RSOR 4. RSQR 5. RPRQ 6. RPRQ Q P 3 x O 4 x x 3 x S R Angles which correspond to the equal sides of an isosceles triangle are equal. © T Madas

P, Q, R and S are points on the circumference of a circle, centre at point O, with PR being a diameter. ROSQ = x and RQPR = 3 x [all angles measured in degrees] Express in terms of x, where appropriate, the following angles: 1. ROSR 2. RORS 3. RSOR 4. RSQR 5. RPRQ 6. RPRQ Q P 3 x O 4 x 180 – 8 x R x 3 x S Angles in a triangle add up to 180° © T Madas

P, Q, R and S are points on the circumference of a circle, centre at point O, with PR being a diameter. ROSQ = x and RQPR = 3 x [all angles measured in degrees] Express in terms of x, where appropriate, the following angles: 1. ROSR 2. RORS 3. RSOR 4. RSQR 5. RPRQ 6. RPRQ Q P 3 x 90 – 4 x O 4 x 180 – 8 x x 3 x S R The inscribed angle is always half the size of a central angle which corresponds to the same arc. © T Madas

P, Q, R and S are points on the circumference of a circle, centre at point O, with PR being a diameter. ROSQ = x and RQPR = 3 x [all angles measured in degrees] Express in terms of x, where appropriate, the following angles: 1. ROSR 2. RORS 3. RSOR 4. RSQR 5. RPRQ 6. RPRQ Q 90 – 4 x 90 P 3 x O 4 x 180 – 8 x R x 3 x S An inscribed angle which corresponds to a diameter is always a right angle. © T Madas

P, Q, R and S are points on the circumference of a circle, centre at point O, with PR being a diameter. ROSQ = x and RQPR = 3 x [all angles measured in degrees] Express in terms of x, where appropriate, the following angles: 1. ROSR 2. RORS 3. RSOR 4. RSQR 5. RPRQ 6. RPRQ Q 90 – 4 x 90 180 – 3 x – 4 x – (90 – 4 x ) = 180 – 3 x – 4 x – 90 + 4 x P 3 x O 4 x 180 – 8 x R = 90 – 3 x x 3 x S Angles in a triangle add up to 180° © T Madas

A regular enneagon (9 -sided polygon) ABCDEFGHI is shown below. Calculate RICE. I A ● for the circle circumscribing H ● ● ● B G O C ● the enneagon: RIOE is a central angle RICE is an inscribed angle both angles correspond to the same arc. hence RICE is half of RIOE ● The central angle of a regular F D E enneagon is given by: 360° ÷ 9 = 40° ● RIOE = 160° ● hence RICE = 80° © T Madas