Circle Theorems (10A) – AlamandaMaths

[fullwidth background_color=”” background_image=”” background_parallax=”none” enable_mobile=”no” parallax_speed=”0.3″ background_repeat=”no-repeat” background_position=”left top” video_url=”” video_aspect_ratio=”16:9″ video_webm=”” video_mp4=”” video_ogv=”” video_preview_image=”” overlay_color=”” overlay_opacity=”0.5″ video_mute=”yes” video_loop=”yes” fade=”no” border_size=”0px” border_color=”” border_style=”” padding_top=”20″ padding_bottom=”20″ padding_left=”0″ padding_right=”0″ hundred_percent=”no” equal_height_columns=”no” hide_on_mobile=”no” menu_anchor=”” class=”” id=””][title size=”2″ content_align=”left” style_type=”none” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Prove and apply angle and chord properties of circles (VCMMG366)
[/title][fusion_text]LO: To prove and apply angle and chord properties of circles

Know:

The properties of a circle
Vocabulary of a circle, such as arc, centre, circumference, radius, diameter, chord, sector and segment.

Understand:

That circles have certain angular properties.

Do:

I can prove and apply angle and chord properities in circles.

[/fusion_text][/fullwidth][title size=”1″ content_align=”left” style_type=”none” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Angle Properties in Circles[/title][fullwidth background_color=”” background_image=”” background_parallax=”none” enable_mobile=”no” parallax_speed=”0.3″ background_repeat=”no-repeat” background_position=”left top” video_url=”” video_aspect_ratio=”16:9″ video_webm=”” video_mp4=”” video_ogv=”” video_preview_image=”” overlay_color=”” overlay_opacity=”0.5″ video_mute=”yes” video_loop=”yes” fade=”no” border_size=”0px” border_color=”” border_style=”” padding_top=”20″ padding_bottom=”20″ padding_left=”0″ padding_right=”0″ hundred_percent=”no” equal_height_columns=”no” hide_on_mobile=”no” menu_anchor=”” class=”” id=””][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

Inscribed angle is an angle created from any 2 points on the circumference of a circle meeting on a 3rd point on the circumference. The central angle is an angle created from any 2 points on the circumference of a circle meeting at the centre of the circle.

The inscribed angle is always half the size of the central angle.

[/fusion_text][button link=”http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/circles2hirev3.shtml” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]BBC Bitesize[/button][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

An angle inscribed in a semicircle is always a right angle.

[/fusion_text][button link=”http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/circles2hirev5.shtml” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]BBC Bitesize[/button][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

Angles in the same segment (same 2 starting points on the circumference) are always equal.

[/fusion_text][button link=”http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/circles2hirev4.shtml” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]BBC Bitesize[/button][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

A cyclic quadrilateral is a quadrilateral (4-sided shape) whose vertices all touch the circumference of a circle.

Opposite angles in a cyclic quadrilateral add up to 180 degrees or is supplementary.

[/fusion_text][button link=”http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/circles2hirev6.shtml” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]BBC Bitesize[/button][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

[/fusion_text][button link=”http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/circles2hirev6.shtml” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]BBC Bitesize[/button][/one_half][title size=”1″ content_align=”left” style_type=”none” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Chord Properties in Circles[/title][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

A chord is a straight line from one point on the circumference of a circle to another.

[/fusion_text][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

If a chord is bisected (split evenly) by the radius of a circle, then both sides of the chord are equal in length.

It also forms a 90 degree angle between the chord and the radius.

[/fusion_text][button link=”http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/circles2hirev7.shtml” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]BBC Bitesize[/button][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

Chords that are equal in length are equidistant (same distance) from the circle centre.

When two chords intersect inside a circle, this divides each chord into two line segments in which the product of the lengths of the line segments for both chords is the same.

[/fusion_text][/one_half][title size=”1″ content_align=”left” style_type=”none” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Tangent and Secants[/title][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

A secant is a line that cuts a circle twice.

A tangent is a line that touches the circumference of a circle at one point only.

[/fusion_text][button link=”http://www.bbc.co.uk/schools/gcsebitesize/maths/geometry/circles2hirev8.shtml” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]BBC Bitesize[/button][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][imageframe lightbox=”no” lightbox_image=”” style_type=”none” hover_type=”none” bordercolor=”” bordersize=”0px” borderradius=”0″ stylecolor=”” align=”none” link=”” linktarget=”_self” animation_type=”0″ animation_direction=”down” animation_speed=”0.1″ hide_on_mobile=”no” class=”” id=””] [/imageframe][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][fusion_text]

A tangent that touches a radius of the circle always creates a 90 degree angle (right angle).

If two tangents intersect outside of a circle, the distances along the tangent from the intersection to the circumference of the circle are equal.

If two secants intersect outside a circle, the product of the entire secant length by the external secant length will be the same.

If a tangent and a secant intersect outside a circle, the product of the entire secant length by the external secant length will be equal to the square of the tangent length.

[/fusion_text][/one_half][/fullwidth][title size=”1″ content_align=”left” style_type=”none” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Circle Properties Videos[/title][fullwidth background_color=”” background_image=”” background_parallax=”none” enable_mobile=”no” parallax_speed=”0.3″ background_repeat=”no-repeat” background_position=”left top” video_url=”” video_aspect_ratio=”16:9″ video_webm=”” video_mp4=”” video_ogv=”” video_preview_image=”” overlay_color=”” overlay_opacity=”0.5″ video_mute=”yes” video_loop=”yes” fade=”no” border_size=”0px” border_color=”” border_style=”” padding_top=”20″ padding_bottom=”20″ padding_left=”0″ padding_right=”0″ hundred_percent=”no” equal_height_columns=”no” hide_on_mobile=”no” menu_anchor=”” class=”” id=””][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][title size=”1″ content_align=”left” style_type=”default” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Inscribed Angles Videos[/title][youtube id=”https://www.youtube.com/watch?v=MyzGVbCHh5M” width=”600″ height=”350″ autoplay=”no” api_params=”” class=””][/youtube][youtube id=”https://www.youtube.com/watch?v=CHe457F9uLg” width=”600″ height=”350″ autoplay=”no” api_params=”” class=””][/youtube][youtube id=”https://www.youtube.com/watch?v=4GpQ267bRgg” width=”600″ height=”350″ autoplay=”no” api_params=”” class=””][/youtube][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][title size=”1″ content_align=”left” style_type=”default” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Circle Theorem[/title][youtube id=”https://www.youtube.com/watch?v=XUus6-9E9sQ” width=”600″ height=”350″ autoplay=”no” api_params=”” class=””][/youtube][/one_half][one_half last=”no” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][title size=”1″ content_align=”left” style_type=”default” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Chord Product Theorem[/title][youtube id=”https://www.youtube.com/watch?v=kS0eJFHDcK8″ width=”600″ height=”350″ autoplay=”no” api_params=”” class=””][/youtube][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][title size=”1″ content_align=”left” style_type=”default” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Secant Product Theorem[/title][youtube id=”https://www.youtube.com/watch?v=hjpiHAiTaWM” width=”600″ height=”350″ autoplay=”no” api_params=”” class=””][/youtube][/one_half][one_half last=”yes” spacing=”yes” center_content=”no” hide_on_mobile=”no” background_color=”” background_image=”” background_repeat=”no-repeat” background_position=”left top” border_position=”all” border_size=”0px” border_color=”” border_style=”” padding=”” margin_top=”” margin_bottom=”” animation_type=”” animation_direction=”” animation_speed=”0.1″ class=”” id=””][title size=”1″ content_align=”left” style_type=”default” sep_color=”” margin_top=”” margin_bottom=”” class=”” id=””]Tangent Product Theorem[/title][youtube id=”https://www.youtube.com/watch?v=-CSL7I_ffLk” width=”600″ height=”350″ autoplay=”no” api_params=”” class=””][/youtube][/one_half][button link=”http://www.alamandamaths.com/” color=”default” size=”” type=”” shape=”” target=”_self” title=”” gradient_colors=”|” gradient_hover_colors=”|” accent_color=”” accent_hover_color=”” bevel_color=”” border_width=”1px” icon=”” icon_position=”left” icon_divider=”no” modal=”” animation_type=”0″ animation_direction=”left” animation_speed=”1″ alignment=”” class=”” id=””]Next Lesson[/button][/fullwidth]