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paulb203

Member

Registered: 2023-02-24

Posts: 426

Geometry; similar shapes

Why are there two possible values of x?

Here's what I've done so far.

AE(SF)=AD
12(SF)=15
SF=15/12
=5/4

AB(SF)=AC
8(5/4)=AC
AC=10

AC-AB=x
x=10-8
x=2

A hint please, as to my next step

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"The secret of getting ahead is getting started."
Mark Twain

Bob

Administrator

Registered: 2010-06-20

Posts: 10,782

Re: Geometry; similar shapes

That's what I did at first.

But the question doesn't say that ADC is an enlargement of AEB, just that the triangles are similar.

So it could be that ACD is an enlargement of AEB. That leads to a second scale factor and hence answer for x.

Important lesson for any maths question with a diagram: Don't make assumptions because of the way the diagram has been drawn.

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

paulb203

Member

Registered: 2023-02-24

Posts: 426

Re: Geometry; similar shapes

Thank, Bob.

What's the difference between ADC and ACD?

I'm wondering why they aren't two ways of labelling the same thing, like we could call/label a square ABCD, going from, say, top left, clockwise through B,C and D. Or we could call/label the same square DCBA, going from bottom left, anti-clockwise through C,B and A. Is that correct?

What am I missing regards ADC v ACD

They look to me like the same shape, the same triangle, the former being labelled anti-clockwise, the latter clock-wise. But obviously I'm missing something

Or maybe I'm just confused about the very basics of this kind of thing?

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"The secret of getting ahead is getting started."
Mark Twain

Bob

Administrator

Registered: 2010-06-20

Posts: 10,782

Re: Geometry; similar shapes

We have a slightly different way of tackling this question. It comes to the same thing, but my way shows how I got two answers.

When I have two triangles that are similar I write them like this:

AEB
ADC

What this means is that A is common to both; D is the enlargement of E and C is the enlargement of B.

The ratio of a side in the larger to the matching side in the smaller will be equal to the ratio of another pair. (equals the enlargement factor) 

ie.  AC/AB = AD/AE.  This leads to your calculation and hence solves for x.

But that assumes that CD is parallel to BE.  It looks right but the question doesn't specifically say so.

A is obviously common but maybe angle ACD = AEB which still leads to similar triangles but with the second answer.

My notation for this case is

ACD
AEB

Now I can pick out the alternative ratios

AC/ AE = AD/AB

and form a second equation for x.

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

paulb203

Member

Registered: 2023-02-24

Posts: 426

Re: Geometry; similar shapes

Thanks, Bob.

“But that assumes that CD is parallel to BE.  It looks right but the question doesn't specifically say so.”
I’ve had another look; it doesn’t actually look parallel.

“A is obviously common but maybe angle ACD = AEB which still leads to similar triangles but with the second answer.”
So E would be the top of the triangle?
Do we rotate the small triangle anti-clockwise until its base is BA?

“My notation for this case is
ACD
AEB”

That makes me think we don’t rotate the small triangle anti-clockwise until its base is BA.
Maybe we flip AEB on its head, as it were, to put E at the top, with A still at the bottom left?
“Now I can pick out the alternative ratios
AC/ AE = AD/AB
and form a second equation for x.”

To find Scale Factor (SF):

(8+x)/(12) = (15)/(8) ?
8x=12
x=12/8
x=3/2
SF=3/2

**

15(3/2)=22.5
22.5-8=14.5
x=14.5 ?

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"The secret of getting ahead is getting started."
Mark Twain

Bob

Administrator

Registered: 2010-06-20

Posts: 10,782

Re: Geometry; similar shapes

That's what I got!

Bob

---

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

paulb203

Member

Registered: 2023-02-24

Posts: 426

Re: Geometry; similar shapes

Cheers, Bob.

I've now looked at the answer and Maths Genie agrees

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"The secret of getting ahead is getting started."
Mark Twain