Explanation:
We are using the
Pythagorean Theorem, which says:
For a right triangle,
$$
a^2 + b^2 = c^2
$$
where $c$ is the hypotenuse (the longest side, opposite the right angle), and $a$ and $b$ are the other two sides.
Let’s solve each problem one by one.
---
Problem 1:
TV is 32 inches long (horizontal side), diagonal is 40 inches (hypotenuse). Find width (vertical side).
Let width = $w$.
Then:
$$
32^2 + w^2 = 40^2 \\
1024 + w^2 = 1600 \\
w^2 = 1600 - 1024 = 576 \\
w = \sqrt{576} = 24
$$
So width =
24.0 inches (already to nearest tenth).
✔ Check: $32^2 + 24^2 = 1024 + 576 = 1600 = 40^2$ ✔️
---
Problem 2:
Ramp forms a right triangle: height = 20 in (vertical), ramp length = 40 in (hypotenuse), find horizontal distance = $x$.
$$
x^2 + 20^2 = 40^2 \\
x^2 + 400 = 1600 \\
x^2 = 1200 \\
x = \sqrt{1200}
$$
Simplify:
$\sqrt{1200} = \sqrt{400 \cdot 3} = 20\sqrt{3} \approx 20 \times 1.732 = 34.64$
Rounded to nearest tenth:
34.6 inches
✔ Check: $34.6^2 + 20^2 ≈ 1197.16 + 400 = 1597.16$, close to 1600 (small rounding error — okay).
Better to compute more precisely:
$\sqrt{1200} = 34.641016...$ →
34.6
---
Problem 3:
Tree = 13 ft tall (vertical), Daniel is 4 ft from base (horizontal), kite is at top of tree (so vertical side = 13 ft), distance from Daniel to kite is hypotenuse.
Let distance = $d$:
$$
d^2 = 4^2 + 13^2 = 16 + 169 = 185 \\
d = \sqrt{185} \approx 13.601...
$$
Rounded to nearest tenth:
13.6 feet
✔ Check: $13.6^2 = 184.96$, very close to 185.
---
Problem 4:
Rope = 25 ft (hypotenuse), horizontal distance from mast base to deck attachment = 15 ft. Find height of mast = $h$.
$$
h^2 + 15^2 = 25^2 \\
h^2 + 225 = 625 \\
h^2 = 400 \\
h = \sqrt{400} = 20
$$
So height =
20.0 feet
✔ Check: $15^2 + 20^2 = 225 + 400 = 625 = 25^2$ ✔️
---
Problem 5:
Ladder = 10 ft (hypotenuse), foot is 5 ft from wall (horizontal), find how high it reaches = $h$.
$$
h^2 + 5^2 = 10^2 \\
h^2 + 25 = 100 \\
h^2 = 75 \\
h = \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3} \approx 5 \times 1.732 = 8.66
$$
Rounded to nearest tenth:
8.7 feet
✔ Check: $8.7^2 = 75.69$, a bit high; actual $\sqrt{75} = 8.660254...$ → rounds to
8.7
---
Final Answer:
1) 24.0
2) 34.6
3) 13.6
4) 20.0
5) 8.7
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem word problem worksheet.
