Question

main ideas/questions
triangle proportionality theorem
notes/examples

• triangle proportionality theorem:

if a line is parallel to one side of a triangle and intersects the other two sides, then it divides the sides into segments of proportional lengths.
if de || ac, then bd/be = da/ec.
triangle diagram with points b, d, e, a, c; d on ba, e on bc, de parallel to ac

• converse of the triangle proportionality theorem:

if bd/be = da/ec, then de || ac.
examples
directions: find the value of x.

1. triangle diagram with sides 14, 12, x, 15; work shown: 14/12 = x/15 → 12x=210 → x=17.5
2. triangle diagram with sides 18, x, 56, 21; work shown: 18/21 = x/56 → 21x=1008 → x=48
3. triangle diagram with sides 55, 45, x, 36; work shown: 10/45 = x/36 (10=55-45) → 45x=360 → x=8
4. triangle diagram with sides 7, x, 6, 27; work shown (implied proportionality)
5. triangle diagram with sides 30, 25, x+7, 15; work shown: 30/25 = (x+7)/15 → 25(x+7)=450 → 25x+175=450 → 25x=275 → x=11
6. triangle diagram with sides 6, 21, x-1, 3x+1; work implied

© gina wilson (all things algebra®, llc), 2014 - 2018

Explanation:

Let's solve problem 6 using the Triangle Proportionality Theorem.

Step 1: Identify the proportion

The Triangle Proportionality Theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Here, the two parallel lines create proportional segments. So we can set up the proportion: $\frac{6}{21}=\frac{x - 1}{3x + 1}$.

Step 2: Cross - multiply

Cross - multiplying gives us $6(3x + 1)=21(x - 1)$.

Step 3: Expand both sides

Expanding the left side: $6\times3x+6\times1 = 18x + 6$.
Expanding the right side: $21\times x-21\times1=21x - 21$.
So the equation becomes $18x + 6=21x - 21$.

Step 4: Solve for x

Subtract $18x$ from both sides: $6 = 21x-18x - 21$.
Simplify the right side: $6 = 3x-21$.
Add 21 to both sides: $6 + 21=3x$.
$27 = 3x$.
Divide both sides by 3: $x=\frac{27}{3}=9$.

Answer:

$x = 9$