Distance Formula Practice worksheet with coordinate geometry problems.

Let’s solve each problem step by step using the distance formula:

> Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

We’ll calculate each one carefully.

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Problem 1: Find the distance between (-12, 1) and (12, -1)

x₁ = -12, y₁ = 1
x₂ = 12, y₂ = -1

Δx = 12 - (-12) = 24
Δy = -1 - 1 = -2

Distance = √[24² + (-2)²] = √[576 + 4] = √580

√580 ≈ ?

Let’s compute:
24² = 576 → so √580 is just a bit more than 24.
Try 24.08² = ?

24.08 × 24.08:

= (24 + 0.08)² = 24² + 2×24×0.08 + 0.08²
= 576 + 3.84 + 0.0064 = 579.8464 → very close to 580!

So √580 ≈ 24.08

✔ Correct answer: C. 24.08

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Problem 2: Find the distance between (-5, -8) and (-1, -16)

x₁ = -5, y₁ = -8
x₂ = -1, y₂ = -16

Δx = -1 - (-5) = 4
Δy = -16 - (-8) = -8

Distance = √[4² + (-8)²] = [16 + 64] = 80

√80 = √(16×5) = 4√5 ≈ 4 × 2.236 = 8.944

Rounded to one decimal? The options are:
A. 10.3
B. 9.8
C. 11
D. 8.9 ← closest!

Wait — let me check again.

Actually, √80 ≈ 8.944 → rounds to 8.9 if we go to one decimal place.

But wait — maybe they want exact value? No, options are decimals.

Option D is 8.9 — that’s correct rounding.

But let me double-check calculation:

(-1 - (-5)) = 4 → squared = 16
(-16 - (-8)) = -8 → squared = 64
Total = 80 → sqrt(80) = 8.944... → yes, 8.9 when rounded to one decimal.

✔ Correct answer: D. 8.9

Wait — hold on! Let me check option B: 9.8 — that’s too big. Option A: 10.3 — way off. C: 11 — no. So D is best.

But actually — sometimes in these problems, they might expect you to round differently? Let me see:

If I use calculator: √80 = 8.94427191 → so to one decimal: 8.9

Yes.

✔ Final for #2: D. 8.9

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Problem 3: Find the distance between (2, -75) and (10, 235)

x₁ = 2, y₁ = -75
x₂ = 10, y₂ = 235

Δx = 10 - 2 = 8
Δy = 235 - (-75) = 235 + 75 = 310

Distance = √[8² + 310²] = [64 + 96100] = √96164

Now compute √96164.

Note: 310² = 96100
So 96164 is just a bit more than 310.

Try 310.1² = ?

310.1 × 310.1 = (310 + 0.1)² = 310² + 2×310×0.1 + 0.1²
= 96100 + 62 + 0.01 = 96162.01 → very close to 96164!

Difference: 96164 - 96162.01 = 1.99

So try 310.11²:

= (310.1 + 0.01)² = 310.1² + 2×310.1×0.01 + 0.0001
≈ 96162.01 + 6.202 + 0.0001 ≈ 96168.212 → too big.

Wait — better approach:

We have √96164.

Since 310.1² = 96162.01
Then 310.1 + δ → where δ is small.

Let’s do linear approx:

f(x) = x², f'(x) = 2x

At x=310.1, f(x)=96162.01
We need f(x) = 96164 → difference = 1.99

δ ≈ 1.99 / (2×310.1) ≈ 1.99 / 620.2 ≈ 0.0032

So √96164 ≈ 310.1 + 0.0032 ≈ 310.1032

So approximately 310.1

Looking at options:

A. 308.6
B. 310.1 ← matches
C. 305.5
D. 315.3

✔ Correct answer: B. 310.1

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Problem 4: Find the distance between (-4, 5) and (7, 18)

x₁ = -4, y₁ = 5
x₂ = 7, y₂ = 18

Δx = 7 - (-4) = 11
Δy = 18 - 5 = 13

Distance = √[11² + 13²] = √[121 + 169] = √290

√290 ≈ ?

17² = 289 → so √290 ≈ 17.029...

Check: 17.03² = ?

17.03 × 17.03 = (17 + 0.03)² = 289 + 2×17×0.03 + 0.0009 = 289 + 1.02 + 0.0009 = 290.0209 → very close!

So √290 ≈ 17.03

Options:

A. 16.23
B. 17.03 ← perfect
C. 16.08
D. 15.96

✔ Correct answer: B. 17.03

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Problem 5: Find the distance between (-3, -11) and (8, -42)

x₁ = -3, y₁ = -11
x₂ = 8, y₂ = -42

Δx = 8 - (-3) = 11
Δy = -42 - (-11) = -31

Distance = √[11² + (-31)²] = √[121 + 961] = √1082

Compute √1082.

32² = 1024
33² = 1089 → so between 32 and 33.

1089 - 1082 = 7 → so closer to 33.

Try 32.9²:

32.9 × 32.9 = (33 - 0.1)² = 1089 - 2×33×0.1 + 0.01 = 1089 - 6.6 + 0.01 = 1082.41 → a bit high.

Try 32.89²:

= (32.9 - 0.01)² = 32.9² - 2×32.9×0.01 + 0.0001
≈ 1082.41 - 0.658 + 0.0001 ≈ 1081.7521 → too low.

Wait — better:

We know 32.9² = 1082.41
We want 1082 → difference = -0.41

Derivative: 2×32.9 = 65.8

δ ≈ -0.41 / 65.8 ≈ -0.0062

So √1082 ≈ 32.9 - 0.0062 ≈ 32.8938

So approximately 32.89, which rounds to 32.9

Options:

A. 32.7
B. 32.9 ← matches
C. 32.3
D. 32.1

✔ Correct answer: B. 32.9

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Problem 6: Find the distance between J and K.

From graph:

Point J: looks like (-3, 4) — because it's 3 left, 4 up from origin.

Point K: looks like (3, 2) — 3 right, 2 up.

Confirming:

J: x=-3, y=4
K: x=3, y=2

Δx = 3 - (-3) = 6
Δy = 2 - 4 = -2

Distance = √[6² + (-2)²] = √[36 + 4] = √40

√40 = √(4×10) = 2√10 ≈ 2 × 3.162 = 6.324

But wait — the options aren’t shown in your image for question 6. You cut off after “Find the distance between J and K.” and there’s a grid.

In your original upload, under question 6, it says:

“Find the distance between J and K.” and then shows a coordinate plane with points J and K plotted.

Assuming standard grid:

J is at (-3, 4)
K is at (3, 2)

As above: Δx=6, Δy=-2 → distance = √(36+4)=√40≈6.32

But since options aren't visible, perhaps in the full quiz, options are given? In your text, you didn’t include them.

Wait — looking back at your original message:

After question 5, it says:

“6. [graph] Find the distance between J and K.”

And then below that, probably options were meant to be there but got cut off.

However, in many such quizzes, for this setup, common options might be around 6.3 or similar.

But since you didn’t provide options for #6, and the instruction is to solve the problems accurately, I’ll assume we need to compute it.

But wait — in your initial prompt, you said “Solve the problem accurately” and listed 6 questions, but only showed partial for #6.

Perhaps in the actual quiz, options are present. Since you didn’t include them, I’ll compute the exact value.

But let me recheck coordinates from graph description:

You wrote: “J .” near top left, “K .” near middle right.

On grid: J is at (-3, 4), K at (3, 2) — yes.

So distance = √[(3 - (-3))² + (2 - 4)²] = √[6² + (-2)²] = √[36+4] = √40 = 2√10 ≈ 6.324555...

If forced to choose from typical multiple choice, likely 6.3 or 6.32, but since no options given, perhaps the question expects exact form? But previous ones are decimal.

Wait — in your original image transcription, after question 6, it just ends. Probably a formatting issue.

Given that, and since all other answers are decimal approximations, I’ll go with 6.32 if needed, but without options, hard to say.

But perhaps in the context, since it’s a practice sheet, and previous answers match calculations, for consistency, I’ll note that.

Actually — looking back at your user input, you included the entire quiz text, and for question 6, it stops after “Find the distance between J and K.” with no options listed. That suggests maybe it was cut off.

To proceed, I’ll assume the coordinates are as read: J(-3,4), K(3,2), distance = √40 ≈ 6.32

But since no options, perhaps skip? No — must answer.

Alternatively, maybe in the graph, K is at (2,2)? Let me think.

Standard interpretation: if grid has x from -3 to 3, y from -4 to 4.

J is at column -3, row 4 → (-3,4)

K is at column 3, row 2 → (3,2) — yes.

Perhaps the intended answer is 40, but since others are decimal, likely 6.3 or 6.32.

But to match format, and since 6.32 is common, but let’s see — in some systems, they might have 6.3 as option.

Since you didn’t provide options for #6, and my task is to solve accurately, I’ll state the calculated value.

But for the sake of completing, and assuming typical options, I’ll go with 6.32 — but wait, let's calculate precisely.

√40 = 6.32455532... → so to two decimals: 6.32

If the quiz had options like:

A. 6.3
B. 6.32
C. 6.4
etc., then B.

But since not specified, and to be consistent with other answers which are to two decimals (like 24.08, 17.03, etc.), I'll use 6.32

However, upon second thought — in problem 2, they used one decimal (8.9), problem 5 used one decimal (32.9), so mixed.

Problem 1: 24.08 (two decimals)
Problem 2: 8.9 (one)
Problem 3: 310.1 (one)
Problem 4: 17.03 (two)
Problem 5: 32.9 (one)

So inconsistent. For #6, since √40 is irrational, likely they expect approximation.

Perhaps in the actual quiz, options are given. Since you didn't include, I'll compute and box the most reasonable.

Another idea: maybe K is at (2,2)? Let me visualize.

If K is at (2,2), then Δx = 2 - (-3) = 5, Δy = 2-4= -2, distance = √(25+4)=√29≈5.385 — not nice.

Or if K is at (3,1)? Then Δy=1-4=-3, Δx=6, distance=√(36+9)=√45=6.708 — not matching.

Best bet: J(-3,4), K(3,2) → √40 ≈ 6.32

I think for accuracy, I'll put 6.32, but since no options, perhaps the system expects us to know.

Wait — in your initial request, you said "solve the problem accurately" and listed 6 questions, so I must provide answer for #6.

Perhaps in the graph, the points are clear. Another way: count squares.

From J to K: move 6 units right, 2 units down → so hypotenuse of right triangle with legs 6 and 2.

Same as before.

I think it's safe to go with 6.32

But to match the pattern, and since problem 4 had 17.03 for 290, similarly here √40=6.324→6.32

So I'll use that.

Final Answer for #6: 6.32 — but since no options, and to be precise, I'll write it.

Actually, let's confirm with calculation:

6.32^2 = 6.32 * 6.32

6*6=36, 6*0.32=1.92, doubled is 3.84, plus 0.32^2=0.1024 → total 36 + 3.84 + 0.1024 = 39.9424 — close to 40.

6.33^2 = 6.33*6.33 = (6.3+0.03)^2 = 39.69 + 2*6.3*0.03 + 0.0009 = 39.69 + 0.378 + 0.0009 = 40.0689 — a bit over.

So 6.32^2=39.9424, 6.33^2=40.0689, target 40.

Interpolate: 40 - 39.9424 = 0.0576
Difference between 6.33^2 and 6.32^2 = 0.1265

So fraction: 0.0576 / 0.1265 ≈ 0.455

So add 0.01 * 0.455 = 0.00455 → so 6.32 + 0.00455 ≈ 6.32455 — same as before.

So to two decimals, it's 6.32, as 6.324 rounds to 6.32? No — 6.324 to two decimals is 6.32, since third digit is 4<5.

6.324 — the digit in hundredths place is 2, thousandths is 4, so yes, rounds to 6.32.

Some might argue 6.32 is fine.

In many textbooks, they leave it as √40 or approximate to 6.3.

But given the context, I'll go with 6.32

However, upon checking online or standard practices, for such problems, often they expect the exact radical or simplified, but here all others are decimal, so decimal it is.

Perhaps the answer is 2√10, but unlikely for this level.

I think for the purpose, I'll box 6.32 for #6.

But to be thorough, let's see if there's a mistake in coordinates.

Another possibility: in the graph, J might be at (-3,3)? But you said "J ." and in description, typically it's at (-3,4).

Perhaps K is at (2,2)? Let's calculate that.

If K(2,2), J(-3,4): Δx=5, Δy=-2, dist=√(25+4)=√29≈5.385 — not nice.

If K(3,1): Δx=6, Δy=-3, dist=√(36+9)=√45=3√5≈6.708

If K(4,2): Δx=7, Δy=-2, dist=√(49+4)=√53≈7.28

None seem better.

Perhaps J is at (-2,4)? Then with K(3,2): Δx=5, Δy=-2, same as above.

I think (-3,4) and (3,2) is correct.

So distance = √40 = 2√10 ≈ 6.3246 → 6.32 when rounded to two decimal places.

So I'll go with that.

Now, compiling all answers:

1. C. 24.08
2. D. 8.9
3. B. 310.1
4. B. 17.03
5. B. 32.9
6. 6.32 (assuming no options, but if options were given, select accordingly)

But since for #6 no options provided in your query, and to complete, I'll state the numerical value.

Perhaps in the actual quiz, options are like:

A. 6.3
B. 6.32
C. 6.4
D. 6.5

Then B.

I think it's fine.

Final Answers:

For the student, I'll list them clearly.

──────────────────────────────────────

Final Answer:
1. C
2. D
3. B
4. B
5. B
6. 6.32

Note: For question 6, since options were not provided in the query, the calculated distance is approximately 6.32 based on coordinates J(-3,4) and K(3,2). If specific options are available in the actual quiz, select the closest match.