Question

Prove that a line that divides two sides of a triangle proportionally is parallel to the third side. Be sure to create and name the appropriate geometric figures.

Answer 1

Triagle ADE similar to Triangle ABC -> Angle D = Angle B -> the 2 lines DE and BC are parallel

Answer 2

Given: In triangle ABC, D and E are two points, so respectively AD/DB = AE/EC.

Prove: DE is parallel to BC.

In triangle ABC, given, AD/DB = AE/EC. Let us assume that in triangle ABC, the point F is an intersect on the side AC. So we can see that AD/DB = AF/FC.

Simplify, AE/EC = AF/FC

Add 1 on both sides, (AE/EC) + 1 = (AF/FC) + 1

(AE+EC)/EC = (AF+FC)/FC

AC/EC = AC/FC

EC = FC

From the work above, we can say that the points E and F coincide on AC, and DF coincides with DE. Since DF is parallel to BC, DE is also parallel BC.

Answer 3

where does point F come from?

Answer 4

the triangles have to be similar for the lines to be parallel.
So show that the triangles are similar, I suppose...

Answer 5

Allen can complete 1/16 of the project per hour
Brianne can complete 1/18 of the project per hour
Charles can complete 1/x

Together they can accomplish 1/16 + 1/18 + 1/X per hour.
we need a common denominator, which is 288X
=
18x+16x+288288x


Choose t as the time it takes them all, so 1/t is the hourly rate.

1/t =
18x+16x+288288x



or t =
288x18x+16x+288


Once the project is completed, Jocelyn will have t, so she will only need to solve for x in this equation to find out the rate 1/x for Charles working by himself (assuming he gets good training!)

Answer 6

Given:\[\frac{ AD }{ DB}=\frac{ AE }{ EC } \Rightarrow \frac{ DB }{ AD}=\frac{ EC }{ AC } \]

add 1 to both sides... (AD/AD on the left and AC/AC on the right).

simplify and realize that AD+DB = AB and that AE + EC = AC and you should be able to find that\[\frac{ AB }{AC }=\frac{ AD }{ AE }\]

Answer 7

Since both triangle share the common angle, A, then you have traingle ADE is similar triangle ABC

Once you have that you can show that the corresponding angles ADE and ABC are congruent, implying that DE and BC are parallel.

Answer 8

Is there a postulate/theorem etc, that can be used after proving the triangles similar?

Answer 9

once you have the traingles are similar, then the corresponding angles are congruent. if you have two lines cut by a transversal whos corresponding angles are congruent, then the lines are parallel.
https://www.mathsisfun.com/geometry/parallel-lines.html

Answer 10

sorry, triangles

Answer 11

thank you!!

Answer 12

http://hotmath.com/hotmath_help/topics/corresponding-angles-postulate.html

this one is better if you have to name a postulate/theorem