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Pythagorean theorem and distance formula

15 questions

9th - 9th Grade

Mathematics

This Grade 9 mathematics worksheet develops fluency with the Pythagorean theorem and the distance formula. Students solve for missing side lengths in equations involving variable expressions, identify correct radical or whole-number solutions, and use the distance formula to find lengths between coordinate points. The 15-item set includes seven fill-in-the-blank questions, one math-response problem, five multiple-choice questions, and two multiple-select questions. Students also round a distance to the nearest hundredth and apply the formulas to a fire-escape platform problem. The worksheet is useful for introducing, reinforcing, or checking students’ ability to connect right-triangle reasoning with coordinate geometry. Several items require students to solve for a variable in a distance relationship, including cases with more than one valid value, so teachers can assess both procedural accuracy and attention to all possible solutions. This free printable worksheet includes a complete answer key for efficient review, independent practice, or assessment.

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Worksheets

Pythagorean theorem and distance formula

Total questions: 15

Name
Class
Date
1.

Use the pythagorean theorem to find the missing values of x.

a = 8, b = x, and c = x+ 2

[Just enter the numerical answer].

(a)  

2.

Use the pythagorean theorem to find the missing values of x.

a = x, b = x+41, and c = 85

[Just enter the numerical answer].

(a)  

3.

Use the pythagorean theorem to find the missing values of x.

a = x-32, b = x-1, and c = x

[Just enter the numerical answer].

Hint: Side measurements cannot be negative. Plug in and check for each side.

(a)  

4.

Find the distance between the two platforms of the fire escape.

Round to the nearest hundredths place.

5.

Solve for the missing variable

a)

10

b)

100

c)

√10

d)

√62

6.

Solve for the missing variable

a)

√365

b)

√27

c)

3√3

d)

5

7.

Use the Distance Formula to find the distance between: (−7, 5) and (2, −7)\left(-7,\ 5\right)\ and\ \left(2,\ -7\right)




(a)  

8.

Use the Distance Formula to find the distance between:
(1, 5) and ( 3, 8)\left(1,\ 5\right)\ and\ \left(\ 3,\ 8\right)  

Round to nearest hundredth.



(a)  

9.

Use the Distance Formula to find the distance between: (−2, −1) and (6, 5)\left(-2,\ -1\right)\ and\ \left(6,\ 5\right)


(a)  

10.

Use the Distance Formula to find the distance between:
(7, 8) and (2, −4)\left(7,\ 8\right)\ and\ \left(2,\ -4\right)  





(a)  

11.

Find the value(s) of "a" if the points with the given vertices are the indicated distance apart.

(-2, -5) and (a, 7) ; distance = 13

[2 answers].

a)

3

b)

-7

c)

-3

d)

7

e)

2

12.

Find the value(s) of "a" if the points with the given vertices are the indicated distance apart.

(8, -2) and (5, a) ; distance = 3

[2 answers].

a)

-2

b)

2

c)

3

d)

-3

13.

Find the value(s) of "a" if the points with the given vertices are the indicated distance apart.

(4, a) and (1, 6) ; distance = 5

[2 answers].

a)

2

b)

10

c)

8

d)

-2

e)

-10

14.

The distance between point B: (-6,-4) and Point D: (-3, -7) written correctly is:

a)

(−7+3)2 + (−4+6)2\sqrt{\left(-7+3\right)^2\ +\ \left(-4+6\right)^2}

b)

(−3+6)2 + (−7+4)2\sqrt{\left(-3+6\right)^2\ +\ \left(-7+4\right)^2}

c)

(−3−6)2 − (−7−4)2\sqrt{\left(-3-6\right)^2\ -\ \left(-7-4\right)^2}

d)

(3+6)2 − (7+4)2\sqrt{\left(3+6\right)^2\ -\ \left(7+4\right)^2}

15.

The distance between point K: (1,2) and Point L: (9,5) written correctly is:

a)

(9−5)2 + (2−1)2\sqrt{\left(9-5\right)^2\ +\ \left(2-1\right)^2}

b)

(9−1)2 + (5−2)2\sqrt{\left(9-1\right)^2\ +\ \left(5-2\right)^2}

c)

(9−5)2 − (2−1)2\sqrt{\left(9-5\right)^2\ -\ \left(2-1\right)^2}

d)

(9−2)2 − (5−1)2\sqrt{\left(9-2\right)^2\ -\ \left(5-1\right)^2}

Answer Key

Pythagorean theorem and distance formula

Total questions: 15

1.
15
2.
36
3.
41
4.

14.1314.13

5.
a)

10

6.
c)

3√3

7.
15
8.
3.61
9.
10
10.
13
11.
a)

3

, b)

-7

12.
a)

-2

13.
a)

2

, b)

10

14.
b)

(−3+6)2 + (−7+4)2\sqrt{\left(-3+6\right)^2\ +\ \left(-7+4\right)^2}

15.
b)

(9−1)2 + (5−2)2\sqrt{\left(9-1\right)^2\ +\ \left(5-2\right)^2}

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FAQs

What does this Pythagorean theorem and distance formula worksheet cover?

This Grade 9 worksheet asks students to use the Pythagorean theorem to solve for missing side variables and use the distance formula to calculate lengths between points on a coordinate plane. Its 15 items combine fill-in-the-blank, math-response, multiple-choice, and multiple-select formats, including rounding distances and identifying all valid values of a variable.

How can I use this worksheet to teach the Pythagorean theorem and distance formula?

Introduce the worksheet by having students identify the two coordinate differences or the legs and hypotenuse before substituting into either formula. Model one variable-expression problem and one distance-formula problem, then ask students to explain why a side length cannot be negative and how rounding changes the final response. Use the fire-escape distance problem as a transition from symbolic work to application.

What mistakes should I watch for when students complete this worksheet?

Watch for students assigning the hypotenuse incorrectly, forgetting to square or square-root both sides, or accepting a value that makes an expression for a side length negative. In the coordinate items, students may subtract coordinates inconsistently, omit the square root, round before the final step, or select only one valid value when a multiple-select question requires two.

How do I assign and grade this Pythagorean theorem and distance formula worksheet?

On Wayground, assign this worksheet as a printable PDF or as a digital quiz. The complete answer key supports grading the fill-in-the-blank, math-response, multiple-choice, and multiple-select items, and printable submissions can be graded by scanning student work with the Wayground for Teachers app. The printable format can also support schools that are reducing screen time.

How can I differentiate this worksheet for my students?

In the worksheet’s Advanced Settings, adjust font spacing or choose a larger font size to make the formulas, coordinate pairs, and answer choices easier to read. You can also enable a dyslexia-friendly font or translate the worksheet into another language while keeping the Pythagorean theorem and distance-formula practice intact.

Where can I find more free worksheets like this on Wayground?

Wayground offers a broad collection of free printable worksheets and practice problems across subjects, with downloadable PDFs and answer keys to help students master essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

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