Grade 8 Workbook — Chapter 10: The Pythagorean Theorem

SOL 8.MG.4 · Companion to Textbook Chapter 10

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 132.

PAGE 1 — Chapter opener

Chapter 10 · The Pythagorean Theorem

Standard 8.MG.4

In this chapter you will:

Verify the Pythagorean Theorem yourself, with diagrams, cut-out squares, and a ruler
Name the hypotenuse and the legs of a right triangle drawn in any orientation
Find a missing side, whether it is the hypotenuse or a leg
Decide whether a triangle is a right triangle from its three side lengths alone
Solve ladder, ramp, screen, field, and grid-distance problems

Words to know: right triangle · right angle · leg · hypotenuse · opposite · Pythagorean Theorem · perfect square · square root · irrational number · rational approximation · converse · Pythagorean triple

Rounding rule for this chapter: give the exact answer in radical form, such as $\sqrt{34}$ , and then a rational approximation to the nearest hundredth, such as $\approx 5.83$ . Use $\approx$ , never $=$ , once you round.

The theorem, to copy out once in your own handwriting:

$a^2 + b^2 = c^2 \qquad \text{where } c \text{ is the hypotenuse}$

WRITING SPACE: 1 in tall, full width

PAGE 2 — The parts of a right triangle

10.1 The Parts of a Right Triangle

FIGURE: fig1-parts-of-a-right-triangle.png (full width)

Fill in the blanks.

The two sides that form the right angle are the ____________.

The side opposite the right angle is the ____________.

The hypotenuse is always the ____________ side of a right triangle.

The hypotenuse is the only side that does not touch the ____________ vertex.

In the equation $a^2 + b^2 = c^2$ , the letter ______ must stand for the hypotenuse.

A right triangle has legs $9$ cm and $12$ cm and a third side of $15$ cm.

Side opposite the right angle: _______________

PAGE 3 — Orientation does not matter

The Same Triangle, Four Ways

FIGURE: fig2-four-orientations.png (full width)

In every picture the right angle is at $P$ .

Figure	Legs	Hypotenuse
2. (a)
3. (b)
4. (c)
5. (d)

(continued) How did you know, without measuring?
True or false: the hypotenuse is always the side at the bottom of the picture. ______

Explain: _______________________________________________

PAGE 4 — Naming parts

Naming Legs and Hypotenuses

Right triangle $DEF$ has its right angle at $E$ .

a) Legs: _______________ b) Hypotenuse: _______________
A right triangle has sides $7$ , $24$ , and $25$ inches.

Hypotenuse: ______ How do you know? _______________________________
In right triangle $ABC$ the right angle is at $C$ , with $AC = 9$ , $BC = 40$ , $AB = 41$ .

Legs: _______________ Hypotenuse: ______ Longest side: ______
Name the hypotenuse of each triangle.

a) $\triangle PQR$ , right angle at $R$	b) $\triangle XYZ$ , right angle at $X$	c) $\triangle LMN$ , right angle at $M$

Think about it. Could a right triangle have a hypotenuse of $8$ ft and a leg of $10$ ft? ______

Why? _______________________________________________
Explain. Why must the hypotenuse be the longest side? (The two non-right angles add to $90^\circ$ .)
Apply it. A ladder leans against a wall.

Hypotenuse: _______________ Legs: _______________________________
Find the error. A student says, "The vertical side has to be a leg, because legs are the straight up-and-down sides."

What is wrong? _______________________________________________

The rule that always works: _______________________________________

PAGE 5 — Exit ticket 10.1

Exit Ticket 10.1

Right triangle $JKL$ has its right angle at $J$ . Hypotenuse: _______________
A right triangle has sides $20$ , $21$ , $29$ m. Hypotenuse: ______
In figure (c) of fig2-four-orientations.png, the two legs are: _______________
Write a procedure for finding the hypotenuse in a figure drawn in any orientation.

PAGE 6 — Squares on the sides

10.2 Verifying the Pythagorean Theorem

FIGURE: fig3-squares-on-the-sides.png (full width)

For the $3$ - $4$ - $5$ triangle above:

Area of the square on the short leg: ______

Area of the square on the long leg: ______

Sum of those two areas: ______

Area of the square on the hypotenuse: ______

FIGURE: fig4-counting-unit-squares.png (full width)

Count the unit squares.

Smaller squares: ______ and ______ Largest square: ______

Equation: ______________________

PAGE 7 — Verifying by computing and by measuring

Verify It Yourself

Legs $6$ and $8$ , hypotenuse $10$ .

$a^2 + b^2 =$ ______________ $c^2 =$ ______ Verified? ______
Legs $5$ and $12$ , hypotenuse $13$ .

$a^2 + b^2 =$ ______________ $c^2 =$ ______ Verified? ______

FIGURE: fig5-measure-and-check.png (full width)

Measure the third side of the triangle above with a ruler.

Measured length: ______ cm $6^2 + 8^2 =$ ______ (your measurement) $^2 =$ ______

Verified? ______
Why can $a^2$ be read as the area of a square instead of as a multiplication?

PAGE 8 — Verification practice

Checking More Triangles

Verify each. Compute the two sides of the equation separately.

Triangle	$a^2 + b^2$	$c^2$	Verified?
a) legs $9$ , $12$ ; hyp. $15$
b) legs $7$ , $24$ ; hyp. $25$
c) legs $12$ , $16$ ; hyp. $20$
d) legs $10$ , $24$ ; hyp. $26$

Legs $5$ and $12$ , hypotenuse $13$ . Areas of the three squares: ______, ______, ______

Equation: ______________________
On grid paper, legs $3$ and $5$ along the grid lines.

Square areas on the legs: ______ and ______ Square on the hypotenuse: ______

Hypotenuse, exact: ______ To the nearest hundredth: ______

GRID SPACE: 3 in tall, full width — 1/4 in grid

PAGE 9 — Measurement, failure, and concrete materials

When the Equation Fails

Draw a right triangle with legs $9$ cm and $12$ cm. Measure the hypotenuse.

GRID SPACE: 3.5 in tall, full width — 1/4 in grid

Measured hypotenuse: ______ cm Expected: ______ cm

Why can a measured check only ever come out approximately equal?
A triangle has sides $4$ , $6$ , $8$ cm.

$4^2 + 6^2 =$ ______ $8^2 =$ ______ Equal? ______

What does that tell you about the triangle? _______________________________
Hands on. Describe how to use a loop of string with $12$ evenly spaced knots plus the square corner of a sheet of paper to verify the theorem for the $3$ - $4$ - $5$ triangle.
Apply it. A carpenter measures $3$ ft along one deck edge and $4$ ft along the other, then measures between the marks.

Distance she should find: ______ ft

If she finds $5$ ft $2$ in. instead, what does that mean? ____________________
Find the error. A student writes $3 + 4 = 7$ , notes $7 \neq 5$ , and says the theorem is false.

Error: _______________________________________________

Correct test: ______________________

PAGE 10 — Exit ticket 10.2

Exit Ticket 10.2

Legs $8$ and $15$ , hypotenuse $17$ . $a^2 + b^2 =$ ______ $c^2 =$ ______ Verified? ______
Legs $9$ and $12$ , hypotenuse $15$ . Square areas: ______, ______, ______

Equation: ______________________
On grid paper, legs $4$ and $4$ .

Area of the square on the hypotenuse: ______ Hypotenuse, exact: ______ Nearest hundredth: ______
What is the difference between verifying the theorem with measurement and proving it?

PAGE 11 — Finding the hypotenuse

10.3 Finding a Missing Side

FIGURE: fig6-find-the-hypotenuse.png (full width)

Case 1 — the hypotenuse is missing: square both legs, ____________, then take the square root.

Case 2 — a leg is missing: square the hypotenuse and the known leg, ____________, then take the square root.

Sanity check: a leg must always come out ____________ than the hypotenuse.

Legs $9$ and $12$ . $c^2 =$ ______ $c =$ ______
Legs $5$ and $12$ . $c^2 =$ ______ $c =$ ______
Legs $7$ and $24$ . $c^2 =$ ______ $c =$ ______

PAGE 12 — Finding a leg

Finding a Leg

FIGURE: fig7-find-a-leg.png (full width)

Hypotenuse $10$ , leg $6$ . $b^2 =$ ______ $b =$ ______
Hypotenuse $26$ , leg $10$ . $b^2 =$ ______ $b =$ ______

When the answer is irrational, give both forms.

FIGURE: fig8-irrational-hypotenuse.png (full width)

Legs $2$ and $6$ . $c^2 =$ ______ Exact: ______ Nearest hundredth: ______
Hypotenuse $9$ , leg $4$ . $b^2 =$ ______ Exact: ______ Nearest hundredth: ______
Hypotenuse $20$ , leg $12$ . $b^2 =$ ______ $b =$ ______

PAGE 13 — Missing sides, mixed

Practice: Hypotenuse or Leg?

Find each hypotenuse.

a) legs $12$ , $16$	b) legs $20$ , $21$	c) legs $9$ , $40$	d) legs $18$ , $24$

Find each missing leg.

a) hyp. $25$ , leg $15$	b) hyp. $50$ , leg $14$	c) hyp. $34$ , leg $16$	d) hyp. $61$ , leg $11$

Legs $5$ m and $5$ m. Exact: ______ Nearest hundredth: ______
Legs $6$ ft and $9$ ft. Exact: ______ Nearest hundredth: ______
Legs $4$ in. and $7$ in. Exact: ______ Nearest hundredth: ______

PAGE 14 — Irrational legs and reasoning

More Missing Sides

Hypotenuse $8$ , leg $5$ . Exact: ______ Nearest hundredth: ______
Hypotenuse $14$ , leg $9$ . Exact: ______ Nearest hundredth: ______
Hypotenuse $15$ , leg $11$ . Exact: ______ Nearest hundredth: ______
Legs $4$ and $6$ .

Exact: ______ Between which two whole numbers? ______ and ______ Nearest hundredth: ______
Explain. Why does the leg case subtract while the hypotenuse case adds?
Find the error. Hypotenuse $13$ , leg $5$ . A student writes $13^2 + 5^2 = 194$ and answers $\approx 13.93$ .

Error: _______________________________________________

The one-second check: _______________________________

Correct answer: ______
Find the error. Legs $6$ and $8$ ; a student answers $c = 14$ .

Error: _______________________________________________

Use the $3$ - $4$ - $5$ triangle to show adding lengths cannot work: ____________________

PAGE 15 — Exit ticket 10.3

Exit Ticket 10.3

Legs $10$ and $24$ . $c =$ ______
Hypotenuse $16$ , leg $9$ . Exact: ______ Nearest hundredth: ______
Legs $3$ and $6$ . Exact: ______ Nearest hundredth: ______
How do you decide whether to add two squares or to subtract one square from another?

PAGE 16 — The converse

10.4 The Converse: Is It a Right Triangle?

FIGURE: fig9-converse-three-triangles.png (full width)

The three-step test.

Circle the ____________ side. That side is ______.
Compute ______________ and ______ separately.
Equal means ____________________. Unequal means ____________________.

Decide. Show the comparison.

#	Sides	$a^2 + b^2$	$c^2$	Right triangle?
61	$9$ , $12$ , $15$
62	$4$ , $6$ , $8$
63	$8$ , $15$ , $17$
64	$5$ , $6$ , $8$
65	$7$ , $24$ , $25$
66	$10$ , $10$ , $15$

PAGE 17 — Converse practice

Testing Three Side Lengths

Decide whether each is a right triangle.

a) $12$ , $16$ , $20$	b) $6$ , $7$ , $10$	c) $20$ , $21$ , $29$	d) $9$ , $12$ , $16$

$14$ , $48$ , $50$ . $a^2 + b^2 =$ ______ $c^2 =$ ______ Right? ______
$13$ , $84$ , $85$ . $a^2 + b^2 =$ ______ $c^2 =$ ______ Right? ______
$11$ , $60$ , $61$ . $a^2 + b^2 =$ ______ $c^2 =$ ______ Right? ______
$30$ , $16$ , $34$ . I used $c =$ ______ because ____________________

$a^2 + b^2 =$ ______ $c^2 =$ ______ Right? ______
$25$ , $24$ , $7$ . I used $c =$ ______ because ____________________

$a^2 + b^2 =$ ______ $c^2 =$ ______ Right? ______
$2$ , $3$ , $4$ . $a^2 + b^2 =$ ______ $c^2 =$ ______ Right? ______

PAGE 18 — Converse reasoning

Why the Longest Side Must Be $c$

Explain. Why must $c$ be the longest side?

Now test $8$ , $15$ , $17$ using $15$ as $c$ : $8^2 + 17^2 =$ ______ $15^2 =$ ______

What wrong conclusion would that give? _______________________________
Apply it. A garden bed has sides $9$ , $12$ , and $16$ ft.

$a^2 + b^2 =$ ______ $c^2 =$ ______ Square corner? ______

What does the direction of the inequality tell you about the largest corner? ____________
Find the error. A student tests $10$ , $26$ , $24$ with $10^2 + 26^2 = 776$ and $24^2 = 576$ , and answers "not right."

Error: _______________________________________________

Correct conclusion: ______________________

PAGE 19 — Exit ticket 10.4

Exit Ticket 10.4

$15$ , $20$ , $25$ — right triangle? ______ Show the comparison: ____________________
$5$ , $7$ , $9$ — right triangle? ______ Show the comparison: ____________________
$18$ , $24$ , $30$ — right triangle? ______ Show the comparison: ____________________
State the converse in your own words, and say why the first step is finding the longest side.

PAGE 20 — Ladders, screens, and grids

10.5 Applying the Theorem in Context

The routine: sketch → label the legs and the hypotenuse → decide which side is unknown → solve → round and write the unit.

FIGURE: fig10-ladder-against-a-wall.png (half width, left)

How high up the wall does the ladder reach?

Which side is unknown, a leg or the hypotenuse? ____________

$h^2 =$ ______ $h =$ ______ ft

FIGURE: fig11-rectangle-diagonal.png (half width, right)

Find the diagonal of the $24$ in. by $10$ in. screen.

$d^2 =$ ______ $d =$ ______ in.

FIGURE: fig12-distance-on-a-grid.png (half width, left)

Find the distance from $A(1, 1)$ to $B(7, 9)$ .

Horizontal leg: ______ Vertical leg: ______ Distance: ______ units

PAGE 21 — Guided applications

More Contexts

A ramp rises $5$ ft over a run of $12$ ft. Sloped surface: ______ ft

SKETCH SPACE: 1.8 in tall, full width
A square garden is $10$ m on a side. Diagonal path, exact: ______ Nearest hundredth: ______ m
A frame is $9$ in. by $12$ in., and its diagonal measures $15$ in.

Corners square? ______ Theorem used: ____________________
A wire runs from the top of a $24$ -ft flagpole to a stake $7$ ft from the base. Wire length: ______ ft
A driver goes $12$ mi north, then $16$ mi east. Straight-line distance: ______ mi

SKETCH SPACE: 1.8 in tall, full width

PAGE 22 — Independent applications

Apply the Theorem

A tabletop is $30$ in. by $40$ in. Diagonal: ______ in.
A kite is on a $26$ -m string, held $10$ m horizontally from the point below the kite. Height: ______ m
Walk $8$ blocks east, $5$ blocks north. Exact: ______ Nearest hundredth: ______ blocks
A field is $6$ m by $9$ m. Diagonal, exact: ______ Nearest hundredth: ______ m
One corner measures $30$ , $40$ , $50$ ft; another measures $20$ , $30$ , $40$ ft.

Corner	$a^2 + b^2$	$c^2$	Square?
$30$ , $40$ , $50$
$20$ , $30$ , $40$

A $20$ -ft ladder must reach a window $16$ ft up. Distance from the wall: ______ ft
A bracket has arms of $7$ in. and $5$ in. joined by a diagonal brace.

Exact: ______ Nearest hundredth: ______ in.
Find the error. A student says a $13$ -ft ladder with its foot $5$ ft from the wall reaches $\sqrt{194} \approx 13.93$ ft.

Error: _______________________________________________

Why is that answer impossible here? _______________________________

Correct height: ______ ft

PAGE 23 — Exit ticket 10.5

Exit Ticket 10.5

A gate is $6$ ft wide and $8$ ft tall. Diagonal brace: ______ ft
A ramp rises $4$ ft over a run of $8$ ft. Exact: ______ Nearest hundredth: ______ ft
A sail has sides $11$ , $15$ , $19$ ft. Square corner? ______ Comparison: ____________________
How do you tell from the wording whether a problem needs the theorem or its converse?

PAGE 24 — Chapter 10 review, part 1

Chapter 10 Review

Part A · Verifying the theorem

Legs $3$ and $4$ . Areas of the three squares: ______, ______, ______

Equation: ______________________
On grid paper, legs $3$ and $5$ .

Squares on the legs: ______ and ______ Square on the hypotenuse: ______

Hypotenuse, exact: ______ Nearest hundredth: ______

GRID SPACE: 2.5 in tall, full width — 1/4 in grid
Legs $20$ and $21$ , hypotenuse $29$ . $a^2 + b^2 =$ ______ $c^2 =$ ______ Verified? ______
Describe how to verify the theorem by drawing a right triangle with legs $9$ cm and $12$ cm and measuring.

Expected measurement: ______ cm Comparison you will make: ____________________
Sides $4$ , $6$ , $8$ : $a^2 + b^2 =$ ______ $c^2 =$ ______

Why does that failure make the theorem worth stating? _______________________
Explain. Verifying on several triangles versus proving for all right triangles:

PAGE 25 — Chapter 10 review, part 2

Chapter 10 Review (continued)

Part B · Hypotenuse and legs in any orientation

FIGURE: fig2-four-orientations.png (full width)

Figure (b): legs _______________ hypotenuse ______
Figure (d): legs _______________ hypotenuse ______
Right triangle $XYZ$ , right angle at $Z$ . Hypotenuse: ______

How did you know? _______________________________
Sides $16$ , $30$ , $34$ . Hypotenuse: ______
Explain. How can a right triangle be drawn with its hypotenuse horizontal along the bottom of the page?

Procedure that works in any orientation: _______________________________
Find the error. A student names the bottom side of a tilted triangle as the hypotenuse.

Error: _______________________________________________

What should the student look for? _______________________________

PAGE 26 — Chapter 10 review, part 3

Chapter 10 Review (continued)

Part C · Finding a missing side

Legs $12$ and $35$ . $c =$ ______
Legs $14$ and $48$ . $c =$ ______
Hypotenuse $30$ , leg $18$ . Other leg: ______
Hypotenuse $41$ , leg $9$ . Other leg: ______
Legs $2$ and $5$ . Exact: ______ Nearest hundredth: ______
Legs $5$ and $7$ . Exact: ______ Nearest hundredth: ______
Hypotenuse $11$ , leg $6$ . Exact: ______ Nearest hundredth: ______
Hypotenuse $13$ , leg $6$ . Exact: ______ Nearest hundredth: ______

PAGE 27 — Chapter 10 review, part 4

Chapter 10 Review (continued)

Part D · Is it a right triangle?

#	Sides	$a^2 + b^2$	$c^2$	Right triangle?
121	$10$ , $24$ , $26$
122	$5$ , $6$ , $8$
123	$9$ , $40$ , $41$
124	$12$ , $16$ , $21$

Sides $24$ , $7$ , $25$ . I used $c =$ ______ because ____________________

Right triangle? ______
Explain. What goes wrong if $c$ is not the longest side? Use $8$ , $15$ , $17$ .

PAGE 28 — Chapter 10 review, part 5

Chapter 10 Review (continued)

Part E · Applications

A $17$ -ft ladder has its foot $8$ ft from a wall. Height reached: ______ ft
A screen is $20$ in. by $21$ in. Diagonal: ______ in.
Distance from $C(2, 1)$ to $D(10, 7)$ : ______ units

Right triangle used — legs: ______ and ______

GRID SPACE: 3 in tall, full width — 1/4 in grid, axes to 11
A square patio is $12$ ft on a side. Diagonal, exact: ______ Nearest hundredth: ______ ft
A foundation corner measures $18$ , $24$ , $30$ ft. Square? ______ Theorem used: ____________________
A garden is $9$ m by $12$ m, with a path along the diagonal.

Diagonal: ______ m Along two sides: ______ m The diagonal is ______ m shorter.

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Radicals: set every radical as a real radical sign with the bar over the whole radicand, never as sqrt(34). Students must be able to see where the radicand ends.
Figure widths: fig3, fig4, fig5, fig9, and fig2 are full width. fig10, fig11, and fig12 sit at half width with their questions beside them.
Print scale matters on page 7. fig5-measure-and-check.png is a measurement activity: place it so the printed legs are true to a centimeter ruler, or add the note "measure in units of the tick marks shown" so the check still works at any scale.
Grid spaces: use a real quarter-inch grid, not a shaded box, wherever GRID SPACE is called for. Items 27, 28, 102, and 129 all depend on students counting squares.
Sketch spaces: plain white boxes with a hairline border. The routine on page 20 asks for a sketch first, so leave room for one on every context page.
Two blanks for irrational answers. Every item that can come out irrational shows two blanks, "Exact" and "Nearest hundredth". Keep both on one line so students never merge them.
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow.

Grade 8 Workbook — Chapter 10: The Pythagorean Theorem

PAGE 1 — Chapter opener

Chapter 10 · The Pythagorean Theorem

PAGE 2 — The parts of a right triangle

10.1 The Parts of a Right Triangle

PAGE 3 — Orientation does not matter

The Same Triangle, Four Ways

PAGE 4 — Naming parts

Naming Legs and Hypotenuses

PAGE 5 — Exit ticket 10.1

Exit Ticket 10.1

PAGE 6 — Squares on the sides

10.2 Verifying the Pythagorean Theorem

PAGE 7 — Verifying by computing and by measuring

Verify It Yourself

PAGE 8 — Verification practice

Checking More Triangles

PAGE 9 — Measurement, failure, and concrete materials

When the Equation Fails

PAGE 10 — Exit ticket 10.2

Exit Ticket 10.2

PAGE 11 — Finding the hypotenuse

10.3 Finding a Missing Side

PAGE 12 — Finding a leg

Finding a Leg

PAGE 13 — Missing sides, mixed

Practice: Hypotenuse or Leg?

PAGE 14 — Irrational legs and reasoning

More Missing Sides

PAGE 15 — Exit ticket 10.3

Exit Ticket 10.3

PAGE 16 — The converse

10.4 The Converse: Is It a Right Triangle?

PAGE 17 — Converse practice

Testing Three Side Lengths

PAGE 18 — Converse reasoning

Why the Longest Side Must Be cc

PAGE 19 — Exit ticket 10.4

Exit Ticket 10.4

PAGE 20 — Ladders, screens, and grids

10.5 Applying the Theorem in Context

PAGE 21 — Guided applications

More Contexts

PAGE 22 — Independent applications

Apply the Theorem

PAGE 23 — Exit ticket 10.5

Exit Ticket 10.5

PAGE 24 — Chapter 10 review, part 1

Chapter 10 Review

Part A · Verifying the theorem

PAGE 25 — Chapter 10 review, part 2

Chapter 10 Review (continued)

Part B · Hypotenuse and legs in any orientation

PAGE 26 — Chapter 10 review, part 3

Chapter 10 Review (continued)

Part C · Finding a missing side

PAGE 27 — Chapter 10 review, part 4

Chapter 10 Review (continued)

Part D · Is it a right triangle?

PAGE 28 — Chapter 10 review, part 5

Chapter 10 Review (continued)

Part E · Applications

Canva production notes

Why the Longest Side Must Be $c$