Math Is Fun Forum

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

some squares and a triangle...

Let ABC be a triangle. We construct squares ABST and ACUV with centers O_1 and O_2, respectively, as shown. Let M be the midpoint of \overline{BC}.

unitsize(0.8 cm);pair A, B, C, M, S, T, U, V;pair[] O;A = (3,3);B = (0,0);C = (4,0);S = rotate(90,B)*(A);T = A + S - B;V = ro...

(a) Prove that \overline{BV} and \overline{CT} are equal in length and perpendicular.

(b) Prove that \overline{O_1 M} and \overline{O_2 M} are equal in length and perpendicular.

thanks

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some squares and a triangle...

hi,

I'm not getting a sensible diagram for this.

If As (3,3) B is (0,0) and C is (4,0) then (0,1) and (0,2) cannot be the centre of those squares.

Any chance you could post a picture please ?

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: some squares and a triangle...

aha here it is:

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Good. You can read.

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: some squares and a triangle...

Yes Im a full member!!!!!! YEEESSSS!!!!

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some squares and a triangle...

Congratulations on becoming a full member.

Now for part a.

angle BAV = BAC + 90 = angle TAC

in triangles BAV and TAC that angle is equal, and you should be able to find two pairs of equal sides, so they are congruent, (SAS)

So TC = BV

Imagine rotating triangle TAC through 90 degrees, anticlockwise, around point A.

T rotates to B and C rotates to V, so one triangle becomes the other.

So TC must also rotate 90 to become BV, hence they are perpendicular.

part b

Consider triangles CVB and C02M.

M bisects BC and O2 bisects CV, so these triangles are similar and BV is parallel to MO2 and half its length

In the same way O1M is parallel to TC and half its length.

Part b follows directly from this.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

championmathgirl

Member

Registered: 2015-06-01

Posts: 20

Re: some squares and a triangle...

Imagine rotating triangle TAC through 90 degrees, anticlockwise, around point A.

T rotates to B and C rotates to V, so one triangle becomes the other.

So TC must also rotate 90 to become BV, hence they are perpendicular.

Bob

How do we know that you have to rotate it 90 degrees?

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Girls can be just as good as boys at math. We just need to get the same encouragement.

RandomPieKevin

Member

Registered: 2015-07-02

Posts: 29

Re: some squares and a triangle...

deleted

Last edited by RandomPieKevin (2015-08-16 07:23:44)

RandomPieKevin

Member

Registered: 2015-07-02

Posts: 29

Re: some squares and a triangle...

Can someone help with part b? Thanks.

RandomPieKevin

Member

Registered: 2015-07-02

Posts: 29

Re: some squares and a triangle...

Can someone help me prove that

? Thanks.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: some squares and a triangle...

I thought I had done this in post 5.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

ET ag

Member

Registered: 2017-02-22

Posts: 10

Re: some squares and a triangle...

Imagine rotating triangle TAC through 90 degrees, anticlockwise, around point A.

T rotates to B and C rotates to V, so one triangle becomes the other.

So TC must also rotate 90 to become BV, hence they are perpendicular.

Bob

How do we know that you have to rotate it 90 degrees?

Well, if we wanted to rotate point

anti-clockwise

about point

, it would end up at point

This is because

is

thus

must "travel" that angle to reach point

.

(Sorry if the LaTeX doesn't seem very neat, I tried to make it "proper" using the correct commands on this database. Quite ironic how LaTeX is supposed to make math look neater when in fact it makes it look quite messy on this website, if you ask me).

(Funny how I have edited this message like 20 times and I have only sat at my desktop for about eight minutes).

From,
ET ag

Last edited by ET ag (2017-02-24 10:32:54)

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If a second was a minute, and a minute was an hour, how many hours would be in one day? (There are multiple answers and they are debatable)