Question

A solid has faces that consist of 5 congruent squares and 4 congruent equilateral triangles. The solid has 9 vertices. How many edges does the solid have? (no image)
A.
16 edges
B.
18 edges
C.
20 edges
D.
22 edges

Answer 1

Use\[v-e+f=2\]where v is no. of vertices, e is no. of edges, f is no of faces.

Answer 2

so 9- ? + 5 = 2 ?

Answer 3

Figure has 5 square faces PLUS 4 triangular faces, so 9 faces in total.

Answer 4

ut oh, I'm confused.

Answer 5

What don't you understand?

Answer 6

oh wait, 9+9 is 18

Answer 7

So 18 edges, I think :o

Answer 8

Not quite. The equation is given above, and you know that f = 9 and v = 9. Are you able to solve for e?

Answer 9

I think so, because E is what I'm suppose to find out

Answer 10

That's right. Go ahead. What do you get?

Answer 11

20 edges?

Answer 12

OK. Let's take it step by step. The equation is\[v-e+f=2\]and you need to solve for e. I don't like the -e on the left hand side, so my first step would be to get of it by adding e to both sides. Can you do that?

Answer 13

so e will cancel out? v+f=2

Answer 14

But to maintain equality you have to add e to BOTH sides of the equation. So you have to fix up the right hand side.

Answer 15

I don't do too well with just variables :o

Answer 16

It's OK. Work with them just as you would work with numbers.

Answer 17

What do you get when you add e to BOTH sides of the equation?

Answer 18

v+ f(e)= 2 ?

Answer 19

No. Let's write it out. Adding e to both sides of the equation we get\[v-e+f+e=2+e\]See how we did that?

Answer 20

ooh yes okay

Answer 21

Now, we can simplify the left hand side by combining the -e and the +e. They combine to give zero, and we don't have to write +0. Some people say that they cancel each other out. Understand?

Answer 22

Yep yep, that part makes sense

Answer 23

So now we have\[v+f=2+e\]To solve for e we have to get rid of the 2 on the right hand side. How will we do that?

Answer 24

we subtract 2 from both sides, but I'm not too sure how that will look

Answer 25

That's right. Let's write it out just like before.\[v+f-2=2+e-2\]Does that make sense?

Answer 26

ohh okay, I understand

Answer 27

Almost there. Now, simplify the right hand side. What will the equation look like?

Answer 28

we subtract 2 again right?

Answer 29

Yup. There's a 2 and a -2 on the right hand side. They 'cancel each other out'.

Answer 30

so v + f - 2 = e ?

Answer 31

Perfect. Well done. Now just plug in the values for v and f and calculate the value of e. What do you get?

Answer 32

9+ 9 ? Because earlier you said 4 faces and 5 faces, which is 9 :o

Answer 33

But you're forgetting something. v=9 and f=9 so we have\[e=v+f-2\]\[e=9+9-2\]\[e=?\]

Answer 34

18-2 = 16

Answer 35

Yayyyy!