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Geometry 9-12 / Week 06 / Tuesday
2/6
Week 06 · Similar Triangles and Proof

Tuesday

The side-splitter theorem
// Shadows, side-splitters and the altitude
⏱ about 20 min

Tuesday: The Side-Splitter Theorem

Wren lays a triangular brace on the bench and a thin strip across it, parallel to the bottom edge. "The cross piece, DE."

"It cuts the two slanted sides," Comet says. "AD is 6 and DB is 4. AE is 9. What is EC?"

"What do you notice about the small triangle ADE and the whole triangle ABC?"

Comet looks. "Same top angle. And because DE is parallel to BC, the angles at D and B match too. AA again."

Nova projects the small triangle sliding down onto the big one, corner A staying put. "Would you like a hint? Compare AB to AD."

"Ten to six. So AC is nine times ten over six: fifteen. EC is six."

Wren checks it with the tape. "Six. Our designer, prove it for every brace."

The theorem

Side-splitter theorem: a line parallel to one side of a triangle divides the other two sides proportionally.

For the brace: DE is parallel to BC, so AD to DB equals AE to EC. Here 6 to 4 and 9 to 6, both 1.5.

Converse: if a line divides two sides of a triangle proportionally, it is parallel to the third side.

Triangle ADE with sides 9 and 6 cm beside the whole triangle ABC with sides 15 and 10 cm.

Proof

  1. Given: triangle ABC with D on AB and E on AC, and DE parallel to BC.
  2. Angle A is shared by triangles ADE and ABC.
  3. Angle ADE = angle ABC, corresponding angles across the parallel lines DE and BC with transversal AB.
  4. Triangle ADE is similar to triangle ABC by AA, so AB over AD equals AC over AE.
  5. AB is AD + DB and AC is AE + EC. So (AD + DB) over AD equals (AE + EC) over AE.
  6. Each side is 1 plus a fraction: 1 + DB over AD = 1 + EC over AE. Subtract 1: DB over AD = EC over AE.

Flip both fractions and the theorem reads AD to DB equals AE to EC. The converse runs the same steps backward to show the angles match.

The crew's brace numbers

SegmentLength (cm)SegmentLength (cm)
AD6AE9
DB4EC6
AB10AC15
AD ÷ DB1.5AE ÷ EC1.5
USE THE PROPORTION
  • Read the question.
  • Tap your answer.
AD is 6 cm and AB is 10 cm. AE is 9 cm. How long is AC, in centimeters?
AC is 15 cm and AE is 9 cm. How long is EC, in centimeters?
On another brace AD is 6 cm, DB is 4 cm and AE is 9 cm. If the cross piece is parallel, what is EC, in centimeters?
A third brace: AD is 5 cm, AB is 10 cm and AE is 7 cm. The cross piece is parallel to BC. How long is AC, in centimeters?
PARALLEL OR NOT?
  • Read the question.
  • Tap your answer.
On a brace, AD ÷ DB = 1.5 and AE ÷ EC = 1.5. Is DE parallel to BC?
On another brace, AD ÷ DB = 1.5 but AE ÷ EC = 2. Is DE parallel to BC?
A triangle ABC with angles 50°, 60°, ?Triangle ADE has angles of 50° and 60°. What is its third angle, which also belongs to triangle ABC?
The side-splitter theorem needs the cross piece to be ____ to the third side. Type the word.
In the proof, triangles ADE and ABC are similar by which criterion? Type the two letters.
StatementTrue or false?
A line parallel to one side of a triangle divides the other two sides proportionally.?
6 ÷ 4 = 9 ÷ 6?
If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.?
The side-splitter proof used SSS.?
WHY THIS EXERCISEThe theorem and its converse let the crew place every cross piece parallel without a square.
Try it
On paper, draw a large triangle. Use the lines of lined paper to draw a cross piece parallel to the bottom.
Measure the four pieces of the two slanted sides and check the two ratios.

Well proved, designer. Tomorrow is Shop Lab: shadows outside, three times in an afternoon.

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