Geometry Chapter 1 Worksheets | Download Free PDF | Area ... - Free Printable

Let's go through each problem in Lesson 1.1 Practice A step by step, solving and explaining the patterns.

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Part 1: Sketch the next figure in the pattern

#### 1. Dots Pattern
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- First figure: 1 dot
- Second: 3 dots (in a triangle)
- Third: 6 dots (triangle with 3 rows)

This is the triangular number pattern:
- 1 = 1
- 3 = 1 + 2
- 6 = 1 + 2 + 3

So the next should be: 1 + 2 + 3 + 4 = 10 dots, arranged in a triangle with 4 rows.

✔ Answer: The next figure has 10 dots arranged in a triangle with 4 rows.

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#### 2. Triangles with arrows and dots
We have three triangles:

1. Equilateral triangle with:
- Arrow pointing down
- Two dots at top corners
2. Triangle rotated so arrow points right
- Dots now at bottom-left and top-right?
- But look closely: the dots seem to move as the triangle rotates.
3. Third triangle: arrow up, dots at bottom corners?

Let’s analyze rotation:
- First: arrow down → second: arrow right → third: arrow up
- So it appears to be rotating counterclockwise by 90° each time.

Now check the dots: They are always at two corners.

In first: dots at top-left and top-right
Second: dots at bottom-left and top-right? Wait — maybe better to label.

But actually, if you rotate the triangle counterclockwise, the positions of the dots shift accordingly.

Wait — perhaps it's about where the dots are placed relative to the arrow?

Alternatively, think of the arrow direction and dot placement:

- Figure 1: Arrow down → dots on the top two vertices
- Figure 2: Arrow right → dots on the left and top? Not clear.

But observe: the arrow is always pointing outward from the center, and the dots are on two corners.

Looking carefully:

- In all figures, there are two dots, and they appear to be on the same two vertices regardless of rotation.

Wait — no. Actually, the dots are fixed in position relative to the triangle, but the triangle rotates.

So if we assume the triangle is being rotated counterclockwise by 90° each time:

- Fig 1: Arrow down → base down
- Fig 2: Arrow right → base left
- Fig 3: Arrow up → base up

So next should be: Arrow pointing left, and the triangle rotated again.

Also, the dots seem to be on the top-left and top-right in first, then in second, they're on bottom-left and top-right?

Wait — perhaps it's simpler: the dots are always on the two vertices that are not the one where the arrow is pointing?

No — in fig 1, arrow down → point at bottom vertex → dots on top-left and top-right → yes.

Fig 2: arrow right → points at right vertex → dots on top-left and bottom-left? No, dots are at bottom-left and top-right? That doesn't match.

Wait — let's re-express:

Actually, upon closer inspection, the dots are always at the two vertices that are not the tip of the arrow.

- Fig 1: Arrow down → tip at bottom → dots at top-left and top-right ✔
- Fig 2: Arrow right → tip at right → dots at top-left and bottom-left? But in image, dots are at bottom-left and top-right? Hmm.

Wait — perhaps the dots are fixed in space, not on the triangle.

But the triangle is moving.

Alternative idea: the arrow moves clockwise, and the dots rotate around.

But let’s consider this: the triangle is being rotated counterclockwise, and the dots remain on the same physical positions.

But the image shows:

- Fig 1: Arrow down → dots on top two corners
- Fig 2: Arrow right → dots on left and top? No, in fig 2, dots are on bottom-left and top-right? That seems inconsistent.

Wait — actually, the dots are always on the two vertices that are not the one the arrow is pointing to, and the arrow is pointing to a different vertex each time.

But in fig 1: arrow down → bottom vertex → dots on top-left and top-right ✔

Fig 2: arrow right → right vertex → dots on top-left and bottom-left? But in image, dots are at top-left and bottom-left? Let's see:

Wait — in fig 2, the triangle is rotated so that the right side is pointing right, so the right vertex is the tip. Then the other two are top and bottom-left. But in the image, dots are on top-left and bottom-left? That would mean both dots are on the left side.

But in fig 1, dots were on top-left and top-right — so not consistent.

Wait — perhaps the dots are always on the two vertices that are not the tip, and the triangle rotates.

So:
- Fig 1: Arrow down → tip at bottom → dots on top-left and top-right
- Fig 2: Arrow right → tip at right → dots on top-left and bottom-left
- Fig 3: Arrow up → tip at top → dots on top-left and top-right? But in fig 3, dots are on bottom-left and bottom-right

Wait — fig 3: arrow up → tip at top → dots at bottom-left and bottom-right → yes!

So pattern:
- Arrow points to a vertex → dots are on the other two vertices

And the arrow is rotating clockwise:
- Down → Right → Up → Next: Left

So next figure: arrow pointing left, and dots on the top and bottom vertices (i.e., the two not at the left tip).

✔ Answer: Next figure is a triangle with arrow pointing left, and dots on top and bottom vertices.

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#### 3. Shaded rectangles
Three rectangles, each divided into two parts:

1. Left half shaded
2. Top half shaded
3. Right half shaded

Pattern: shading alternates between vertical and horizontal, and shifts.

- Fig 1: vertical, left shaded
- Fig 2: horizontal, top shaded
- Fig 3: vertical, right shaded

So alternating between vertical and horizontal, and within vertical: left → right

Next should be horizontal, and since last horizontal was top, next could be bottom.

So next: horizontal division, bottom shaded

✔ Answer: Rectangle divided horizontally, bottom half shaded

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#### 4. Shapes increasing sides
- Triangle (3 sides)
- Square (4 sides) with X inside
- Pentagon (5 sides) with star-like shape

Each figure increases number of sides: 3 → 4 → 5

Next: Hexagon (6 sides), likely with a more complex internal pattern.

The internal pattern: triangle has nothing? Wait — triangle has no internal lines, square has an X (diagonals), pentagon has a star (pentagram).

So:
- Triangle: no internal lines
- Square: diagonals → X
- Pentagon: pentagram (star)

So next: hexagon with a hexagram (six-pointed star) or similar.

But possibly just the shape increases.

But the internal pattern may be: connect every other vertex?

- Square: connect opposite vertices → X
- Pentagon: connect every other vertex → star
- Hexagon: connect every other vertex → two triangles (Star of David)

So next figure: hexagon with a six-pointed star inside

✔ Answer: Hexagon with a star formed by connecting every other vertex (like Star of David)

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Part 2: Describe a pattern and predict the next number

#### 5. 2, 5, 8, 11, ...
- Add 3 each time: 2 → 5 (+3), 5 → 8 (+3), 8 → 11 (+3)
- Next: 14

✔ Answer: Add 3; next is 14

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#### 6. 27, 9, 3, 1, ...
- Divide by 3: 27 ÷ 3 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1
- Next: 1 ÷ 3 = 1/3

✔ Answer: Divide by 3; next is 1/3

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#### 7. 123, 234, 345, 456, ...
- Each number increases by 111: 123 + 111 = 234, etc.
- Next: 456 + 111 = 567

✔ Answer: Add 111; next is 567

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#### 8. 5, 7, 11, 17, 25, ...
Look at differences:
- 7 - 5 = 2
- 11 - 7 = 4
- 17 - 11 = 6
- 25 - 17 = 8

Differences: +2, +4, +6, +8 → increasing by 2

Next difference: +10 → 25 + 10 = 35

✔ Answer: Add 2, 4, 6, 8, 10... next is 35

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#### 9. 1/2, 2/3, 3/4, 4/5, ...
Numerator: 1, 2, 3, 4 → next: 5
Denominator: 2, 3, 4, 5 → next: 6
So next fraction: 5/6

✔ Answer: Numerator and denominator increase by 1; next is 5/6

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#### 10. 5/4, 4/6, 3/8, 2/10, ...
Numerator: 5, 4, 3, 2 → decreasing by 1 → next: 1
Denominator: 4, 6, 8, 10 → increasing by 2 → next: 12
So next: 1/12

✔ Answer: Numerator decreases by 1, denominator increases by 2; next is 1/12

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#### 11. 4, 1, –2, –5, ...
Differences:
- 1 - 4 = -3
- -2 - 1 = -3
- -5 - (-2) = -3

So subtract 3 each time → next: -5 - 3 = -8

✔ Answer: Subtract 3; next is -8

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#### 12. 1, 4, 9, 16, ...
These are perfect squares:
- 1² = 1
- 2² = 4
- 3² = 9
- 4² = 16
- Next: 5² = 25

✔ Answer: Squares of integers; next is 25

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Part 3: How many squares in the next object?

#### 13.
Figures:
1. One square
2. Cross: 5 squares (center + 4 arms)
3. T-shape? Wait — looks like:
- Fig 1: 1 square
- Fig 2: 5 squares (like a plus sign)
- Fig 3: More complex — looks like a 3x3 grid with some missing?

Wait — let's count:

- Fig 1: 1 square
- Fig 2: 5 squares (center + 4 adjacent)
- Fig 3: Looks like a 3x3 cross? Or more?

Actually, looking at the shapes:

- Fig 1: single square
- Fig 2: cross with 5 squares (one in center, one on each side)
- Fig 3: Larger cross — seems like a plus with arms of length 3?

Wait — actually, it's growing.

Alternatively, look at the number of squares:

- Fig 1: 1
- Fig 2: 5
- Fig 3: ? Count: center, up, down, left, right, and extra on ends?

Wait — in fig 3, it looks like:
- Vertical line: 3 squares (up, middle, down)
- Horizontal line: 3 squares (left, middle, right)
- But middle shared → total: 3 + 3 - 1 = 5? But that’s same as fig 2.

No — fig 3 has more.

Wait — fig 3 has:
- Bottom row: 3 squares
- Middle: 3 squares
- Top: 3 squares
- But arranged in a cross?

Wait — actually, the figures are:

1. One square
2. A cross: 5 squares (central square with one on each side)
3. A larger cross: central square, then 2 squares on each arm?

Yes — so:
- Fig 1: 1 square
- Fig 2: 1 + 4 = 5 squares
- Fig 3: 1 + 8 = 9 squares? But wait — arms are longer?

No — fig 3 has:
- Center
- Up: 2 squares
- Down: 2 squares
- Left: 2 squares
- Right: 2 squares

Wait — but that would be 1 + 8 = 9

But visually, fig 3 has:
- 3 vertical: up, center, down
- 3 horizontal: left, center, right
- But center shared → 3 + 3 - 1 = 5? No.

Wait — look at the image:

Actually, fig 3 has:
- A central square
- One square above, one below
- One to the left, one to the right
- Plus two more on the top and bottom? No.

Wait — the third figure has a central column of 3 squares, and a horizontal bar of 3 squares crossing at center → total: 3 + 3 - 1 = 5? Still 5.

But fig 2 is also 5.

Wait — no: fig 2 is a cross with arms of length 1 → 5 squares

fig 3: arms of length 2 → so:
- Vertical: 3 squares (up, center, down)
- Horizontal: 3 squares (left, center, right)
- Total: 3 + 3 - 1 = 5? Same?

No — that can’t be.

Wait — actually, fig 3 has more squares.

Looking at the image:

- Fig 1: one square
- Fig 2: five squares in a cross (like plus sign)
- Fig 3: looks like a 3x3 grid with the four corners missing? No.

Wait — fig 3 has:
- A central square
- Above it: one square
- Below: one
- Left: one
- Right: one
- And then above the top one: another square? No — no.

Wait — fig 3 has:
- Three squares vertically (top, middle, bottom)
- Three squares horizontally (left, middle, right)
- But middle shared → total: 5

Same as fig 2.

But fig 3 is taller — wait, it has a square on top of the top one? No.

Wait — actually, the third figure has:
- A vertical stack of 3 squares
- A horizontal bar of 3 squares intersecting at the center
- So total: 3 + 3 - 1 = 5

But that’s same as fig 2.

That can’t be.

Wait — perhaps the pattern is:

- Fig 1: 1 square
- Fig 2: 5 squares (cross)
- Fig 3: 9 squares? Like a 3x3 grid?

But it’s not a full 3x3.

Wait — maybe it's building a larger cross.

Another idea: the number of squares increases by 4 each time?

- Fig 1: 1
- Fig 2: 5 → +4
- Fig 3: 9 → +4

So next: 13

But fig 3 has 9? Let’s count:

From the image:

- Fig 1: 1
- Fig 2: 5 (central + four directions)
- Fig 3: central, then two in each direction? So:
- Up: 2 squares
- Down: 2
- Left: 2
- Right: 2
- Center: 1
- Total: 1 + 8 = 9

Yes! So:
- Fig 1: 1 square (arms of length 0? or 1?)
- Better: the length of each arm increases.

Actually:
- Fig 1: 1 square → maybe arm length 0, just center?
- Fig 2: arm length 1 → 1 center + 4 × 1 = 5
- Fig 3: arm length 2 → 1 center + 4 × 2 = 9

So pattern: 1 + 4n, where n is arm length.

So next: arm length 3 → 1 + 4×3 = 13 squares

✔ Answer: 13 squares

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#### 14.
Figures:
- Fig 1: 2 squares (L-shape? or two stacked)
- Fig 2: 5 squares (like a staircase)
- Fig 3: 9 squares?

Wait — count:

- Fig 1: 2 squares (one on top of another)
- Fig 2: 5 squares — looks like a 3-level staircase: 1 + 2 + 2? No.

Wait — fig 1: 2 squares stacked vertically
fig 2: 5 squares — looks like:
- Left: 2 squares
- Middle: 2 squares
- Right: 1 square

Wait — actually, it's a staircase:

- Fig 1: 2 squares (height 2, width 1)
- Fig 2: height 3, width 2? But only 5 squares

Wait — better: look at number of squares:

- Fig 1: 2
- Fig 2: 5
- Fig 3: 9

Check differences: 5 - 2 = 3, 9 - 5 = 4 → not constant

Or look at structure:

- Fig 1: 2 squares (stacked)
- Fig 2: 5 squares — looks like a 2×2 block plus one on top? Or staircase.

Wait — actually, it might be triangular numbers or something else.

Wait — fig 1: 2 squares
fig 2: 5 squares
fig 3: 9 squares

2, 5, 9 → differences: +3, +4 → next +5 → 14?

But let’s see the pattern.

Alternatively, think of layers:

- Fig 1: 2 squares (vertical)
- Fig 2: 5 squares — maybe 2 on left, 2 on right, 1 in middle?
- Fig 3: 9 squares — maybe 3 on left, 3 on right, 3 in middle?

Wait — no.

Wait — fig 1: 2 squares
fig 2: 5 squares
fig 3: 9 squares

Try: 2 = 1 + 1, 5 = 2 + 3, 9 = 3 + 6? Not helpful.

Another idea: the number of squares follows:
- Fig 1: 2 = 1² + 1
- Fig 2: 5 = 2² + 1
- Fig 3: 9 = 3²? 3² = 9 → yes!

But 1² + 1 = 2, 2² + 1 = 5, 3² = 9? Inconsistent.

Wait — 2, 5, 9 → not quadratic.

Wait — 2 = 1×2, 5 = 5, 9 = 3×3

Another approach: look at the shape.

It appears to be a staircase or increasing steps.

- Fig 1: 2 squares (height 2)
- Fig 2: 5 squares — height 3, width 2?
- Fig 3: height 4, width 3?

Wait — perhaps the number of squares is:

- Fig 1: 2
- Fig 2: 5
- Fig 3: 9

Try: 2, 5, 9 → differences: +3, +4 → next +5 → 14

So next: 14

But let’s verify with pattern.

Wait — another possibility: the total number of squares in a "step" pattern.

Alternatively, think of it as:

- Fig 1: 2 squares
- Fig 2: 5 squares
- Fig 3: 9 squares

Then next: 14? (since 2→5→9→14: +3, +4, +5)

So the pattern is adding consecutive integers: +3, +4, +5 → next +6 → 14 + 6 = 20?

Wait — no: sequence is 2, 5, 9, ?, ?

Differences: 3, 4 → so next difference 5 → 9 + 5 = 14

Then next: +6 → 20

But we need only the next one.

But is this supported?

Alternatively, look at the figure:

- Fig 1: 2 squares
- Fig 2: 5 squares — looks like a 3-step staircase: 1+2+2? No.

Wait — fig 1: 2 squares (maybe 1+1)
fig 2: 5 squares — 2+3? Or 1+2+2?

Wait — actually, fig 2 has:
- Bottom row: 3 squares
- Middle: 2 squares
- Top: 0? No.

Wait — better: fig 2 has:
- Left column: 2 squares
- Middle: 2 squares
- Right: 1 square → total 5

fig 3:
- Left: 3
- Middle: 3
- Right: 3 → total 9

So:
- Fig 1: 2 = 1 + 1
- Fig 2: 5 = 2 + 2 + 1
- Fig 3: 9 = 3 + 3 + 3

Not consistent.

Wait — fig 1: 2 squares
fig 2: 5 squares
fig 3: 9 squares

2, 5, 9 → differences: +3, +4 → so next: +5 → 14

And the pattern is: add 3, then 4, then 5, etc.

So next: 14 squares

✔ Answer: 14 squares

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Part 4: Complete the conjecture

#### 15. Product of odd and even number
Examples:
- 3 × 8 = 24 → even
- 6 × 5 = 30 → even
- 11 × 24 = 264 → even
- 102 × 31 = 3162 → even

All products are even.

Why? Because any multiple of an even number is even.

So product of odd and even is always even.

✔ Answer: The product is even

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#### 16. Sum of odd and even number
Examples:
- 17 + 22 = 39 → odd
- 8 + 37 = 45 → odd
- 135 + 48 = 183 → odd
- 94 + 85 = 179 → odd

All sums are odd.

Because: odd + even = odd

✔ Answer: The sum is odd

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✔ Final Answers Summary:

1. 10 dots in a triangle
2. Triangle with arrow pointing left, dots on top and bottom vertices
3. Rectangle with bottom half shaded
4. Hexagon with a six-pointed star (or star inside)
5. 14
6. 1/3
7. 567
8. 35
9. 5/6
10. 1/12
11. -8
12. 25
13. 13 squares
14. 14 squares
15. even
16. odd

Let me know if you'd like a visual sketch explanation for any!