Question

Is the following correct, if so how?

Answer 1

The correct answer is 1/4.

Notice that we have three similar triangles in this figure.

This is the Triangle Midsegment Theorem. Why?

HI is a segment that is the midpoint of AN and AO,

NO is a segment that is the midpoint of AB and AC.

Thus that means that

`hi = (1)/(2) (no)`

`no = (1)/(2) (bc)`

Notice that we are asked for AH to AC.

So we are looking at the two similar triangles, AHI and ABC.

By definition, their sides form a proportion.

So that means that for some number, k,

k(AH)=AB

k(HI)=BC

k(AI)=AC

Notice that

`hi = (1)/(2) (no)`

We also know that

`no = (1)/(2) (bc)`

So

`hi = (1)/(2) ( (1)/(2) bc)`

`hi = (1)/(4) bc`

So that means Triangle AHI is 1/4 of ABC

Thus, the ratio is 1/4.