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Geometry 9-12 / Week 06 / Friday
5/6
Week 06 · Similar Triangles and Proof

Friday

Criteria at work
// Shadows, side-splitters and the altitude
⏱ about 20 min

Friday: Criteria at Work

Comet leans the stepladder against the back wall while a grown-up holds it. "Ladder, wall, floor. A right triangle."

"Foot 1.5 meters out, top 2 meters up," Wren reads. "What do you notice about the ladder's length?"

"Square root of 2.25 plus 4. That is 6.25, so 2.5 meters."

Nova projects a second triangle, half the size, beside it. "Would you like a hint? A smaller ladder at the same slant."

"Similar by AA. Foot 0.75 out, top 1 up, length 1.25," Comet says. "Everything halves."

"Review today," Wren says. "Congruence when things are the same size, similarity when they are not. Next week, the ramp angle and trigonometry."

Which criterion?

You knowCriterionConclusion
three pairs of equal sidesSSScongruent
two sides and the angle between them equalSAScongruent
two angles and the side between them equalASAcongruent
two pairs of equal anglesAAsimilar
three pairs of sides in one ratioSSS for similaritysimilar
a line parallel to one side of a triangleside-splitterthe other sides split proportionally

Congruent triangles have equal sides and angles: the scale factor is 1. Similar triangles share angles and one side ratio.

To find a length, name the similar triangles, match corresponding sides, write one proportion, solve.

To prove a relationship, name the criterion first, then read off the equal parts.

The week in five lines

  • Shadow triangles at one moment are similar by AA, so a stick and two shadows measure a pole.
  • A line parallel to one side of a triangle splits the other two sides proportionally, and the converse holds.
  • The altitude to the hypotenuse makes two triangles similar to the whole.
  • Those similar triangles prove a² + b² = c².
  • Congruence criteria fix size and shape; similarity criteria fix shape only.
MIXED REVIEW
  • Read the question.
  • Tap your answer.
A right triangle with legs 1.5 and 2 m and the longest side marked ?The ladder's foot is 1.5 m from the wall and its top is 2 m up. How long is the ladder, in meters?
A smaller ladder at the same slant reaches 1 m up instead of 2 m. How far out is its foot, in meters?
A stick casts a 0.5 m shadow while a mast casts a 3 m shadow. How tall is the mast, in meters?
A right triangle with legs 1.5 and 2 m and the bottom leg marked ?A 2.5 m ladder reaches 2 m up a wall. How far is its foot from the wall, in meters?
NAME THE CRITERION
  • Read the question.
  • Tap your answer.
Two braces have sides 6, 8, 10 and 9, 12, 15. Which criterion applies?
Two braces have sides 6, 8, 10 and 6, 8, 10. Which criterion applies?
A cross piece on a brace splits one side 2 to 3 and the other side 4 to 6. What can you conclude?
Week reviewTrue or false?
Similar triangles have equal corresponding angles.?
Similar triangles always have equal corresponding sides.?
6 × 6 + 8 × 8 = 10 × 10?
The altitude to the hypotenuse splits a right triangle into two triangles similar to it.?
The shadow method works even if the two shadows are measured an hour apart.?
WHY THIS EXERCISEThese five facts are the toolkit for every length you cannot reach with a tape.
Draw the ladder against the wall as a right triangle, then a half-size similar triangle beside it, all sides labeled.
Try it
Find a right triangle at home: a ramp, a doorstop, a leaning broom. Measure the two legs and compute the third side.
Then measure the third side and compare.

Sharp review, designer. Tomorrow is Shop Day: the family measures something tall by its shadow.

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