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Polygons and similar figures
Two figures are similar when one is a scaled copy of the other: same angles, every side multiplied by the same number. That single number answers missing sides, shadow problems and area questions.
Same shape, every side times the same number
Similar figures have the same angles, and their matching sides are all in the same ratio. That ratio is the scale factor. Find it from one pair of matching sides, then use it for every other side.
Multiply, never add
If one side grows from 8 to 14.4, it was multiplied by 1.8, not increased by 6.4. Every other side is multiplied by 1.8 too.
How to solve them
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1
The quadrilaterals and what makes each one
All four-sided shapes have angles that add up to 360°. They differ in which sides are parallel or equal.
Parallelogram2 pairs ∥opposite sides equal and parallelRectangle4 × 90°a parallelogram with right anglesRhombus4 equal sidesa parallelogram with equal sidesSquarebothfour equal sides and four right anglesTrapezoid1 pair ∥exactly one pair of parallel sidesNext step: Polygon anglesShortcutEvery square is a rectangle and a rhombus
A square has everything a rectangle has (right angles) and everything a rhombus has (equal sides). But a rectangle or a rhombus is not always a square.
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2
Interior angles of a polygon
A polygon with n sides has interior angles that add up to (n − 2) × 180°. In a regular polygon (all sides and angles equal), divide by n for each angle.
Next step: Similar figuresPentagon5 sides
540° in all, 108° each if regular
Hexagon6 sides
720° in all, 120° each if regular
Octagon8 sides
1,080° in all, 135° each if regular
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3
Similar figures: find the scale factor, use it everywhere
Match the sides carefully (shortest with shortest, longest with longest). Divide one pair to get the scale factor, or set up a proportion.
Next step: ShadowsExampleTwo similar triangles: one has sides 4, 6 and 9. The shortest side of the other is 10. Find its other sides.
- Scale factor: 10 ÷ 4 = 2.5.
- 6 × 2.5 = 15 and 9 × 2.5 = 22.5.
Answer 15 and 22.5
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4
Shadow problems are similar triangles
At the same moment, the sun makes every object’s height and shadow form similar right triangles. So height ÷ shadow is the same for everything.
Next step: Area of similar shapesExampleA 6-foot person casts a 4-foot shadow. A tree nearby casts an 18-foot shadow. How tall is the tree?
- 6/4 = h/18.
- h = 18 × 6 ÷ 4 = 108 ÷ 4 = 27.
Answer 27 feet
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5
Area grows with the square of the scale factor
If every side is multiplied by k, the perimeter is multiplied by k but the area by k × k. (Volume, for solids, by k × k × k.)
Next step: CongruentSides × 2Area of 10 becomes
10 × 4 = 40
Sides × 3Area of 10 becomes
10 × 9 = 90
PerimeterSides × 3, perimeter 12 becomes
12 × 3 = 36
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Congruent means identical
Congruent figures are similar with a scale factor of 1: same shape and same size. Every matching side and angle is equal.
SimilarSame angles, sides in the same ratio
Same shape, any size
CongruentSame angles and same sides
Same shape, same size
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Adding instead of multiplying
A side grows from 4 to 6. Another side of 10 becomes:
Wrong10 + 2 = 12
Right10 × 1.5 = 15
Area scaled like a side
Every side doubles. An area of 9 becomes:
Wrong18
Right9 × 4 = 36
Matching the wrong sides
Sides 3, 5 and 6, 10. The ratio:
Wrong3 to 10
Right3 to 6 and 5 to 10: × 2
Every rhombus is a square
Must a rhombus have right angles?
Wrongyes
Rightno: only equal sides
5 questions at test level
Pick an answer, then read why. No timer, no account.
What is the measure of each interior angle of a regular hexagon?
Show the answer
C. The angles add up to (6 − 2) × 180 = 720°, shared by 6 equal angles: 720 ÷ 6 = 120. 108° is a regular pentagon and 135° an octagon. Each angle is 120°.
Two triangles are similar. The smaller has sides 5, 7 and 9. The shortest side of the larger is 15. How long is the longest side of the larger triangle?
Show the answer
B. Scale factor: 15 ÷ 5 = 3. The longest side is 9 × 3 = 27. 19 adds 10 instead of multiplying, and 35 multiplies the middle side by 5 instead of 3. The longest side is 27.
A 5-foot fence post casts a 3-foot shadow. At the same time, a building casts a 24-foot shadow. How tall is the building?
Show the answer
B. Height ÷ shadow is the same for both: 5/3 = h/24, so h = 24 × 5 ÷ 3 = 120 ÷ 3 = 40. 14.4 flips the ratio, 72 multiplies the shadow by 3, and 120 forgets to divide by 3. The building is 40 feet tall.
Two similar rectangles have sides in the ratio 1 : 4. The smaller one has an area of 6 square inches. What is the area of the larger one?
Show the answer
C. Sides × 4 means area × 4 × 4 = 16: 6 × 16 = 96. 24 scales the area like a side, and 48 multiplies by 8. The larger area is 96 square inches.
Which shape always has four right angles and four equal sides?
Show the answer
C. A rectangle always has right angles but not always equal sides; a rhombus has equal sides but not always right angles. Only a square must have both. The answer is A square.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished polygons and similar figures.
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