Question

Which statement can be proved true using the given theorem?

Segment BF = 9
Segment BD = 18
Segment BD = 20
Segment BF = 24

Answer 1

Theorem: A line parallel to one side of a triangle divides the other two proportionately.

In the figure below, segment DE is parallel to segment BC and segment EF is parallel to AB:

Answer 2

I got 24 by cross multiplying and then dividing.

Answer 3

Using the theorem, set up a proportion for BF and a proportion for BD.
Solve the proportions, and find their lengths.
Then choose the correct option.

Answer 4

Which proportion did you set up?

Answer 5

12*30 / 15
360/ 15
24

Answer 6

That is not a proportion.
A proportion is two ratios that are equal to each other.
Also, which length are you solving for, BF or BD?

Answer 7

I am so confused

Answer 8

Let me redraw your triangle to simplify it.
You have two parallel segments, so there are two possibilities with the proportions.
Let me draw a triangle with only one parallel side and explain the theorem to you that way.

Answer 9

Okay thank you.

Answer 10

Here is a triangle.

Answer 11

Now I'll draw a segment parallel to side BC that intersects sides AB and AC.

Answer 12

Segment DE is parallel to side BC.

Answer 13

SO 12 and 15

Answer 14

According to the theorem above, then the split sides have proportional segments.
This is what that means:

\(\dfrac{AD}{DB} = \dfrac{AE}{EC} \)

Answer 15

Okay let me solve it

Answer 16

In each split side, you find the ratio of the two segments of the side.
If you write the ratios in the same order, then the ratios are equal.
That's what it means for the segment of the sides to be proportional.

Answer 17

I got 20.

Answer 18

Now we use this figure for your problem and write only the pertinent segment lengths for now.

Answer 19

We can use our proportion to solve for BD.

Answer 20

\(\dfrac{AD}{BD} = \dfrac{AE}{CE} \)

\(\dfrac{12}{BD} = \dfrac{15}{25} \)

Notice only one unknown, so we can solve for BD.

\(BD = \dfrac{12 \times 25}{15} \)

\(BD = 20\)

Answer 21

I already did that. Thank you though.

Answer 22

You helped me get there!!

Answer 23

That is the answer.
Now out of curiosity, let's look at what BF is.

Answer 24

\(\dfrac{30}{BF} = \dfrac{25}{15} \)

\(BF = \dfrac{30 \times 15}{25} \)

\(BF = 18\)

Answer 25

Thank you!!

Answer 26

BF = 18, and no answer states that, so the correct choice is C.

Answer 27

You're welcome.