Math worksheet for plotting coordinates in the first quadrant to form geometric shapes.

Let's solve the problem step by step.

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Section A: Plot and Connect Coordinates to Form Shapes

We are given three sets of coordinates. We need to:

1. Plot each point on the coordinate grid.
2. Join them in order using a ruler.
3. Identify the shape formed.

We’ll analyze each set one at a time.

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1. First Set:

Coordinates:
(2, 11) → (6, 11) → (6, 5) → (2, 5) → (2, 11)

Let’s examine this path:

- From (2,11) to (6,11): horizontal line, same y-value → moves right 4 units.
- From (6,11) to (6,5): vertical line, same x-value → moves down 6 units.
- From (6,5) to (2,5): horizontal left → same y-value, moves left 4 units.
- From (2,5) back to (2,11): vertical up → same x-value, moves up 6 units.

This forms a rectangle because:
- Opposite sides are equal and parallel.
- All angles are 90°.

✔ Shape: Rectangle

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2. Second Set:

Coordinates:
(9,7) → (11,1) → (7,1) → (9,7)

Plotting:

- (9,7) to (11,1): diagonal down-right
- (11,1) to (7,1): horizontal left along y = 1
- (7,1) to (9,7): diagonal up-right

Let’s check the sides:

- (11,1) to (7,1): horizontal side of length 4
- The other two sides go from (9,7) to (11,1) and (7,1) to (9,7)

Now, observe that:
- Two sides meet at (9,7), forming a "V" shape.
- The base is from (11,1) to (7,1), which is horizontal.
- The two non-base sides connect to the top point (9,7).

So, we have:
- Three points: (9,7), (11,1), (7,1)
- But the path goes: (9,7) → (11,1) → (7,1) → (9,7)

This creates a triangle with:
- Base from (11,1) to (7,1)
- Apex at (9,7)

It’s an isosceles triangle because:
- Distance from (9,7) to (11,1):
√[(11−9)² + (1−7)²] = √[4 + 36] = √40
- Distance from (9,7) to (7,1):
√[(7−9)² + (1−7)²] = √[4 + 36] = √40

So both sides are equal → Isosceles triangle

✔ Shape: Isosceles Triangle

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3. Third Set:

Coordinates:
(5,8) → (12,6) → (15,8) → (12,10) → (5,8)

Let’s plot and analyze:

- (5,8) to (12,6): diagonal down-right
- (12,6) to (15,8): diagonal up-right
- (15,8) to (12,10): diagonal up-left
- (12,10) to (5,8): diagonal down-left

Let’s look at symmetry or possible shapes.

Check if it's a kite or parallelogram?

Try calculating midpoints or side lengths.

But easier: look at the pattern.

Notice:
- (5,8) and (15,8) are at same height (y=8)
- (12,6) and (12,10) are on same vertical line (x=12)

So:
- The figure has two diagonals:
- One from (5,8) to (15,8): horizontal
- One from (12,6) to (12,10): vertical

Wait — but the points are connected in order:
- (5,8) → (12,6) → (15,8) → (12,10) → (5,8)

Let’s sketch mentally:

- From (5,8) to (12,6): down-right
- To (15,8): up-right
- To (12,10): up-left
- Back to (5,8): down-left

This looks like a kite!

A kite has:
- Two pairs of adjacent equal sides
- One diagonal as axis of symmetry

Let’s verify:

Check distances:

1. (5,8) to (12,6):
√[(12−5)² + (6−8)²] = √[49 + 4] = √53

2. (12,6) to (15,8):
√[(15−12)² + (8−6)²] = √[9 + 4] = √13

3. (15,8) to (12,10):
√[(12−15)² + (10−8)²] = √[9 + 4] = √13

4. (12,10) to (5,8):
√[(5−12)² + (8−10)²] = √[49 + 4] = √53

So:
- Sides: √53, √13, √13, √53
- Adjacent sides: (5,8)-(12,6) = √53; (12,6)-(15,8) = √13 → different
- But opposite sides: (5,8)-(12,6) and (15,8)-(12,10) both √53? No — wait.

Wait: the sequence is:
- Side 1: (5,8) to (12,6) = √53
- Side 2: (12,6) to (15,8) = √13
- Side 3: (15,8) to (12,10) = √13
- Side 4: (12,10) to (5,8) = √53

So:
- Side 1 and Side 4: both √53
- Side 2 and Side 3: both √13

So two pairs of adjacent sides equal?

Wait: actually:
- (5,8) to (12,6): √53
- (12,6) to (15,8): √13 → not equal to previous

But:
- (12,6) to (15,8): √13
- (15,8) to (12,10): √13 → equal
- (12,10) to (5,8): √53
- (5,8) to (12,6): √53 → equal

So the two diagonals are:
- Horizontal: from (5,8) to (15,8)? Not directly connected.
- Vertical: from (12,6) to (12,10) — yes, they are connected via (12,6) and (12,10), but not directly.

But notice: the figure has:
- Diagonal from (5,8) to (15,8): horizontal, length 10
- Diagonal from (12,6) to (12,10): vertical, length 4

And the vertices are symmetric about the point (12,8)? Let's see:

- (5,8) and (15,8): symmetric about x=10? No — midpoint is x=(5+15)/2 = 10
- But (12,6) and (12,10): midpoint is (12,8)

Not symmetric about same point.

But let’s think differently.

Actually, this shape is a kite if it has two pairs of adjacent equal sides.

But here:
- (5,8) to (12,6): √53
- (12,6) to (15,8): √13 → not equal
- (15,8) to (12,10): √13
- (12,10) to (5,8): √53

So the opposite sides are equal, but not adjacent.

Wait — no: the sides are:
- AB: (5,8) to (12,6) = √53
- BC: (12,6) to (15,8) = √13
- CD: (15,8) to (12,10) = √13
- DA: (12,10) to (5,8) = √53

So:
- AB = DA = √53
- BC = CD = √13

So it's a kite only if two pairs of adjacent sides are equal.

Here:
- AB and BC: √53 and √13 → not equal
- BC and CD: √13 and √13 → equal
- CD and DA: √13 and √53 → not equal
- DA and AB: √53 and √53 → equal

So only two adjacent sides equal: BC and CD

But AB and DA are equal, but not adjacent.

Wait — actually, the shape is symmetric about the vertical line x=12?

Check:

- (5,8) and (15,8): symmetric about x=10? No.
- But (5,8) to (12,6) and (15,8) to (12,10): are they symmetric?

Let’s compute:

From (5,8) to (12,6): Δx = +7, Δy = -2
From (15,8) to (12,10): Δx = -3, Δy = +2 → not symmetric.

Wait — maybe it's a parallelogram?

Check vector AB and DC:

AB: (12−5, 6−8) = (7, -2)
DC: (12−15, 10−8) = (-3, 2) → not equal

AD: (12−5, 10−8) = (7, 2)
BC: (15−12, 8−6) = (3, 2) → not equal

No.

Wait — let’s plot the points roughly:

- (5,8): left
- (12,6): center-down
- (15,8): right
- (12,10): center-up

So the shape connects:
- Left → center-down → right → center-up → left

This looks like a diamond or kite, but actually, it's a quadrilateral with two diagonals crossing at (12,8)?

The diagonals would be:
- (5,8) to (15,8): horizontal, length 10
- (12,6) to (12,10): vertical, length 4

They intersect at (12,8)

And the four vertices are symmetric about the point (12,8)? Let’s check:

- (5,8): distance from (12,8) is 7 left
- (15,8): 3 right → not symmetric

But wait — perhaps it's a kite with symmetry along the vertical line x=12?

Let’s check if (5,8) and (15,8) are symmetric about x=12:

- x=5 → distance 7 left
- x=15 → distance 3 right → no

Not symmetric.

But notice: (12,6) and (12,10) are on the same vertical line.

Now, the connections:
- (5,8) to (12,6): down-right
- (12,6) to (15,8): up-right
- (15,8) to (12,10): up-left
- (12,10) to (5,8): down-left

This is a quadrilateral with two pairs of equal sides:

- (5,8) to (12,6): √[(7)² + (2)²] = √(49+4)=√53
- (12,10) to (5,8): √[(7)² + (2)²] = √53 → same

- (12,6) to (15,8): √[(3)² + (2)²] = √(9+4)=√13
- (15,8) to (12,10): √[(3)² + (2)²] = √13 → same

So:
- Two opposite sides are equal: (5,8)-(12,6) and (12,10)-(5,8) → both √53
- But they are not adjacent.

Wait — actually, the sides are:
- AB: (5,8) to (12,6) = √53
- BC: (12,6) to (15,8) = √13
- CD: (15,8) to (12,10) = √13
- DA: (12,10) to (5,8) = √53

So:
- AB = DA = √53
- BC = CD = √13

But AB and DA are not adjacent — they are opposite.

Wait — no: AB and DA share vertex A.

AB: A to B
DA: D to A → so yes, they are adjacent at A.

Similarly, BC and CD are adjacent at C.

But AB and DA are not equal in direction.

Wait — actually, the two pairs of adjacent sides:

- At A: AB = √53, AD = √53 → both √53 → equal!
- At C: CB = √13, CD = √13 → equal!

So:
- At vertex A: two sides equal (√53)
- At vertex C: two sides equal (√13)

That's the definition of a kite: two pairs of adjacent equal sides.

Yes! So:
- AB = AD = √53 → adjacent sides at A
- BC = CD = √13 → adjacent sides at C

Therefore, this is a kite.

Also, the diagonals are:
- AC: from (5,8) to (15,8) → horizontal
- BD: from (12,6) to (12,10) → vertical

They cross at (12,8), and one diagonal is perpendicular to the other.

So yes — kite.

✔ Shape: Kite

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✔ Final Answers:

| Coordinates | Name of Shape |
|-----------|----------------|
| (2,11) to (6,11) to (6,5) to (2,5) to (2,11) | Rectangle |
| (9,7) to (11,1) to (7,1) to (9,7) | Isosceles Triangle |
| (5,8) to (12,6) to (15,8) to (12,10) to (5,8) | Kite |

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✔ Summary:

- First shape: Rectangle
- Second shape: Isosceles Triangle
- Third shape: Kite

You can now fill in the table accordingly.