Here are the step-by-step solutions for each problem on the worksheet. We will use the Pythagorean Theorem, which is $a^2 + b^2 = c^2$, where $c$ is the longest side (the hypotenuse).
1) Television Screen Width
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Identify the sides: The diagonal is the longest side ($c = 40$). The length is one leg ($a = 32$). We need to find the width ($b$).
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Set up the equation: $32^2 + b^2 = 40^2$
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Calculate squares: $1024 + b^2 = 1600$
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Isolate $b^2$: Subtract 1024 from 1600.
$1600 - 1024 = 576$
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Find the square root: $\sqrt{576} = 24$
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Answer: The width is 24 inches.
2) Ramp Horizontal Distance
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Identify the sides: The ramp length is the hypotenuse ($c = 40$). The height is one leg ($a = 20$). We need the horizontal distance ($b$).
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Set up the equation: $20^2 + b^2 = 40^2$
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Calculate squares: $400 + b^2 = 1600$
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Isolate $b^2$: Subtract 400 from 1600.
$1600 - 400 = 1200$
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Find the square root: $\sqrt{1200} \approx 34.641$
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Round: Rounding to the nearest tenth gives 34.6.
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Answer: The horizontal distance is 34.6 inches.
3) Kite Distance
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Identify the sides: The tree height is one leg ($a = 13$). The distance from Daniel to the tree is the other leg ($b = 4$). We need the distance to the kite, which is the hypotenuse ($c$).
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Set up the equation: $13^2 + 4^2 = c^2$
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Calculate squares: $169 + 16 = c^2$
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Add them up: $185 = c^2$
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Find the square root: $\sqrt{185} \approx 13.601$
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Round: Rounding to the nearest tenth gives 13.6.
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Answer: The distance is 13.6 feet.
4) Sailboat Mast Height
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Identify the sides: The rope is the hypotenuse ($c = 25$). The distance on the deck is one leg ($b = 15$). We need the mast height ($a$).
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Set up the equation: $a^2 + 15^2 = 25^2$
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Calculate squares: $a^2 + 225 = 625$
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Isolate $a^2$: Subtract 225 from 625.
$625 - 225 = 400$
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Find the square root: $\sqrt{400} = 20$
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Answer: The height of the mast is 20 feet.
5) Ladder Height
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Identify the sides: The ladder is the hypotenuse ($c = 10$). The distance from the wall is one leg ($b = 5$). We need the height on the wall ($a$).
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Set up the equation: $a^2 + 5^2 = 10^2$
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Calculate squares: $a^2 + 25 = 100$
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Isolate $a^2$: Subtract 25 from 100.
$100 - 25 = 75$
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Find the square root: $\sqrt{75} \approx 8.660$
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Round: Rounding to the nearest tenth gives 8.7.
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Answer: The ladder reaches 8.7 feet up the wall.
Final Answer:
1) 24 inches
2) 34.6 inches
3) 13.6 feet
4) 20 feet
5) 8.7 feet
Parent Tip: Review the logic above to help your child master the concept of word problems involving pythagorean theorem.
![Pythagorean Theorem [Video] Formula, Definition, Examples & Proof](/images2/word-problems-involving-pythagorean-theorem/word-problems-involving-pythagorean-theorem-32.jpg)
