Question

Andrea drew the diagram shown below and described it as an equilateral triangle with a square inside it. Point D is the midpoint of segment AC.
The geometry teacher pointed out that the triangle cannot be equilateral since side AC is not equal to AB.
Which statement best explains why the measurements are incorrect?

Answer 1

@jim_thompson5910 help?

Answer 2

Segment AB is 6 + 3 + 6 = 15.
Segment AB is 6 + 9 + 6 = 21.
Segment AB ≈ 6 + 6.7 + 6 ≈ 18.7.
Segment AB ≈ 6 + 10.8 + 6 ≈ 22.8.

Answer 3

Ok let's assume that this is an equilateral triangle (it's actually not, but let's assume it is)

Answer 4

those are the options lol

Answer 5

alright thanks

Answer 6

Point D is the midpoint of AC, so AD = DC

Since AD = 9, this means DC = 9

Answer 7

So if our assumption is correct, then triangle CDG must also be an equilateral triangle since this triangle is similar to triangle ABC

Answer 8

But...if ABC is an equilateral triangle, then

AC = AB

18 = AB

AB = 18

which would mean EF = 6 (since this is the last remaining piece of AB)

Telling us that

Answer 9

* DG = 6

Answer 10

Now if triangle CDG is an equilateral triangle, then DC = DG must be true, but this is clearly not the case since DC = 9 and DG = 6


So triangle CDG is NOT an equilateral triangle


which means triangle ABC is NOT an equilateral triangle as well

Answer 11

Does that argument make sense?

Answer 12

Yes, the argument makes sense. Now I'm trying to use it to find my answer

Answer 13

well unfortunately I wrote all that out before I took a look at the options (turns out they use a completely different way lol)...but fortunately I know how to get the right answer

Answer 14

Oh haha
Well, Im thinking its probably either C or D.. but then again, geometry is NOT my thing so Im probably wrong lol><

Answer 15

Alright here's how you do this

Step 1) Find the length of DE

Use the pythagorean theorem

AE^2 + DE^2 = AD^2

6^2 + DE^2 = 9^2

36 + DE^2 = 81

DE^2 = 81-36

DE^2 = 45

DE = sqrt(45)

DE = 6.708

Since quadrilateral DEFG is a square, we know that

EF = DE

EF = 6.708

---------------------------------------------

So

AB = AE+EF+FB

AB = 6+6.708+6

AB = 18.708 (approximately)


But....


AC = AD + DC

AC = 9+9

AC = 18

So AC does NOT equal AB, so ABC can't be equilateral


So going back to the answer choices, the answer is "Segment AB ≈ 6 + 6.7 + 6 ≈ 18.7"

Answer 16

Thank you SO much! Youre a life saver!

Answer 17

you're welcome