Question

I need some Geometry help. Stuck on this problem. Picture attached. Thank you. Can someone please help??

Answer 1

#40

Answer 2

Seems that part 40 a is pretty easy to answer:
Triangles BMN, MAO, NOC, and MNO are congruent

Triangles MQP, QNR, PRO, and PQR are congruent.

They are all equilateral triangles.

Answer 3

I think triangles
Tri BMN
Tri MAO
Tri NOC
Tri MNO

can be shown to be congruent.

The theorem about the segment joining the midpoints of 2 sides of a triangle being parallel to the third side and having length half the third side is at play here.

Then, that smaller cluster of 4 triangles can be proved congruent, I think.

Answer 4

>> They are all equilateral triangles.

I did not pick up on that. They would have to be equilateral. Good call. @wolf1728

Answer 5

Yes I agree. I think just showing they are all equilateral should be sufficient.
Shucks thanks!

Answer 6

There are 16 of the smaller triangle which comprise the outer equilateral triangle. Agree?

So, would the area of the shaded triangles (small and larger) be 13 times the area of the unshaded triangles? Agree?

Answer 7

Area of the big triangle = 16

Area of shaded triangles = 13
Area of unshaded triangles = 3

So, area of shaded to unshaded = 13/3

Answer 8

By the way you helped me a lot on that last answer.
I was getting all set to compute the area of all the triangles and you made it much simpler by counting the area of the whole triangle as 16, etc.

Answer 9

Is it correct to say that the shaded area is 13 times the area of the unshaded area?
That seems not to work out numerically.

Answer 10

13/16 of the triangle is shaded. 3/16 is unshaded. Just leave it at that for the nonce.

Answer 11

Excellent help guys. Made it easy to understand.