### Geometrical Proofs

Introduction

Intercept theorem

There are 2 ways to apply the intercept theorem

In the first scenario, you will be able to identify a pair of parallel lines from the question

Upon identifying the pair of parallel lines,

we conclude that the ratio of AD:DB is the same as the ratio of AE:EC

In the second scenario, you identify that that the ratio of AD:DB is the same as the ratio of AE:EC

Subsequently you conclude that the lines DE and BC are parallel

Midpoint Theorem

The Midpoint theorem is a specific case of the Intercept theorem when the ratio of the lengths is 1:1

There are 2 ways to apply the Midpoint Theorem

The first way is to use Midpoint Theorem to show that 2 lines are parallel

From the information given in the question, we will be able to identify that D is the midpoint of AB and E is the midpoint of AC

Given that D and E are midpoints, we will be able to conclude that DE is parallel to BC

The second way is to use Midpoint Theorem to show that E is a midpoint

From the information given in the question, we know that DE is parallel to BC and D is a midpoint

Using the Midpoint theorem, we can conclude that E is the midpoint of AC

Question

Given that:

O is centre of circle, AB is diameter, C lies on circle

AD is tangent to circle at A, EC is tangent to circle at C

i) Prove that triangles AEO and CEO are congruent

Strategy: make use of information given in the question

AD is tangent to circle --> angle OAE is 90 degrees

EC is tangent to circle --> angle OCE is 90 degrees

O is centre of circle --> OA and OC are radii of the circle, hence their lengths are equal

From diagram OE is the common length

Using the RHS property, we can show that triangles AEO and CEO are congruent

Intercept theorem

There are 2 ways to apply the intercept theorem

In the first scenario, you will be able to identify a pair of parallel lines from the question

Upon identifying the pair of parallel lines,

we conclude that the ratio of AD:DB is the same as the ratio of AE:EC

In the second scenario, you identify that that the ratio of AD:DB is the same as the ratio of AE:EC

Subsequently you conclude that the lines DE and BC are parallel

Midpoint Theorem

The Midpoint theorem is a specific case of the Intercept theorem when the ratio of the lengths is 1:1

There are 2 ways to apply the Midpoint Theorem

The first way is to use Midpoint Theorem to show that 2 lines are parallel

From the information given in the question, we will be able to identify that D is the midpoint of AB and E is the midpoint of AC

Given that D and E are midpoints, we will be able to conclude that DE is parallel to BC

The second way is to use Midpoint Theorem to show that E is a midpoint

From the information given in the question, we know that DE is parallel to BC and D is a midpoint

Using the Midpoint theorem, we can conclude that E is the midpoint of AC

Question

Given that:

O is centre of circle, AB is diameter, C lies on circle

AD is tangent to circle at A, EC is tangent to circle at C

i) Prove that triangles AEO and CEO are congruent

Strategy: make use of information given in the question

AD is tangent to circle --> angle OAE is 90 degrees

EC is tangent to circle --> angle OCE is 90 degrees

O is centre of circle --> OA and OC are radii of the circle, hence their lengths are equal

From diagram OE is the common length

Using the RHS property, we can show that triangles AEO and CEO are congruent

Hey, this is a great post, thank you so much for sharing. I’m looking forward to coming back to your site for more great information. Keep up the good work! : )

ReplyDeleteHey, this is a great post, thank you so much for sharing. I’m looking forward to coming back to your site for more great information. Keep up the good work! : )

ReplyDeleteThis comment has been removed by the author.

ReplyDeleteThis information is meaningful and magnificent which you have shared here Math tuition. I am impressed by the details that you have shared in this post and It reveals how nicely you understand this subject. I would like to thanks for sharing this article here. Maths Tuition In Adelaide

ReplyDeleteIt is truly a well-researched content and excellent wording. I got so engaged in this material that I couldn’t wait to read. I am impressed with your work and skill. Thanks.Online English Classes

ReplyDeleteNice. I am really impressed with your writing talents and also with the layout on your weblog. Appreciate, Is this a paid subject matter or did you customize it yourself? Either way keep up the nice quality writing, it is rare to peer a nice weblog like this one nowadays. Thank you, check also event management and meeting thank you email

ReplyDeleteI just need to say this is a well-informed article which you have shared here about hoodies. It is an engaging and gainful article for us. Continue imparting this sort of info, Thanks to you. English Resources

ReplyDeleteI just need to say this is a well-informed article which you have shared here about hoodies. It is an engaging and gainful article for us. Continue imparting this sort of info, Thanks to you. Higher score in CELBAN listening module

ReplyDeleteExtremely useful information which you have shared here. This is a great way to enhance knowledge for us, and also helpful for us. Thankful to you for sharing an article like this.Premade ebook covers

ReplyDeleteVery good, This information is essential and informative which you have shared here. Read more info about Find A Tutor. It is beneficial for beginners to develop their knowledge. It is very gainful information. Thanks for share it.

ReplyDeleteYour blog contains lots of valuable data. It is a factual and beneficial article for us.best private college in lucknow Thankful to you for sharing an article like this.

ReplyDeleteI just need to say this is a well-informed article which you have shared here about hoodies.best management college in lucknow It is an engaging and gainful article for us. Continue imparting this sort of info, Thanks to you.

ReplyDeleteI liked your work and, as a result, the manner you presented this content about California dmv online driving course.It is a valuable paper for us. Thank you for sharing this blog with us.

ReplyDeleteYou wrote this post very carefully.customer enablement training The amount of information is stunning and also a gainful article for us. Keep sharing this kind of articles, Thank you.

ReplyDeleteI generally check this kind of article and I found your article which is related to my interest.Cantonese Lessons Hong Kong Genuinely it is good and instructive information. Thankful to you for sharing an article like this.

ReplyDeleteYou have a genuine capacity to compose a substance that is useful for us. You have shared an amazing blog which related to education given by Montessori School In Bangalore thanks for sharing this blog with us.

ReplyDeleteI just need to say this is a well-informed article which you have shared here about hoodies.Spanish Private Tutor Online It is an engaging and gainful article for us. Continue imparting this sort of info, Thanks to you.

ReplyDeleteYour essay is fantastic. Here, you've given very insightful knowledge. Coding Classes Hong Kong The information you have provided us with is genuinely beneficial and significant. Thank you for providing this information.

ReplyDelete