📄 8th Grade Math (Algebra I): Pythagorean Theorem Worksheet
📌 1. True / False
1. The Pythagorean Theorem applies to all types of triangles.
2. The hypotenuse is always the longest side of a right-angled triangle.
3. If the sides of a triangle are 3, 4, and 5 units, it is a right-angled triangle.
4. In the Pythagorean Theorem formula \(a^2 + b^2 = c^2\), 'c' represents one of the legs.
5. The Pythagorean Theorem can be used to find the length of a missing side if two sides of a right triangle are known.
✏️ 2. Fill in the Blanks
1. The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the of the squares of the lengths of the other two sides.
2. The two shorter sides of a right-angled triangle that form the right angle are called .
3. A triangle with side lengths 6, 8, and 10 units is a triangle.
4. The formula for the Pythagorean Theorem is \(a^2 + b^2 = \).
5. If a right triangle has legs of length 5 cm and 12 cm, its hypotenuse is cm long.
🔗 3. Matching
« The longest side of a right-angled triangle, opposite the right angle.
« The two shorter sides of a right-angled triangle that form the right angle.
« An angle that measures exactly 90 degrees.
« A set of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\).
« If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
✍️ 4. Short Answer Questions
1. Explain when the Pythagorean Theorem can be applied to a triangle.
💡 Suggested Answer: The Pythagorean Theorem can be applied only to right-angled triangles. It is used to find the length of an unknown side when the lengths of the other two sides are known.
2. What is a Pythagorean Triple? Provide an example.
💡 Suggested Answer: A Pythagorean Triple is a set of three positive integers \(a, b, c\) such that \(a^2 + b^2 = c^2\). An example is (3, 4, 5) because \(3^2 + 4^2 = 9 + 16 = 25\), and \(5^2 = 25\).
🎯 5. Multiple Choice
1. A right-angled triangle has legs measuring 7 cm and 24 cm. What is the length of its hypotenuse?
2. Which of the following sets of side lengths could form a right-angled triangle?
3. The hypotenuse of a right triangle is 10 inches long, and one leg is 6 inches long. What is the length of the other leg?
📝 6. Open-Ended Questions
1. A ladder is leaning against a wall. The base of the ladder is 5 feet away from the wall, and the ladder reaches 12 feet up the wall. What is the length of the ladder?
💡 Solution Steps:
Let 'a' be the distance from the base of the ladder to the wall (5 feet) and 'b' be the height the ladder reaches up the wall (12 feet). Let 'c' be the length of the ladder (the hypotenuse).
According to the Pythagorean Theorem, \(a^2 + b^2 = c^2\).
Substitute the given values:
\(5^2 + 12^2 = c^2\)
\(25 + 144 = c^2\)
\(169 = c^2\)
To find 'c', take the square root of both sides:
\(c = \sqrt{169}\)
\(c = 13\)
The length of the ladder is 13 feet.
2. A rectangular park is 80 meters long and 60 meters wide. If a person walks diagonally across the park from one corner to the opposite corner, how far do they walk?
💡 Solution Steps:
The length and width of the rectangular park form the legs of a right-angled triangle, and the diagonal walk forms the hypotenuse.
Let 'a' be the width (60 meters) and 'b' be the length (80 meters). Let 'c' be the diagonal distance.
Using the Pythagorean Theorem: \(a^2 + b^2 = c^2\)
Substitute the values:
\(60^2 + 80^2 = c^2\)
\(3600 + 6400 = c^2\)
\(10000 = c^2\)
To find 'c', take the square root of both sides:
\(c = \sqrt{10000}\)
\(c = 100\)
The person walks 100 meters diagonally across the park.
3. A television screen has a diagonal measure of 40 inches. If the height of the screen is 24 inches, what is the width of the screen?
💡 Solution Steps:
In this problem, the diagonal of the television screen is the hypotenuse, and the height and width are the legs of a right-angled triangle.
Let 'c' be the diagonal (40 inches) and 'b' be the height (24 inches). Let 'a' be the unknown width.
Using the Pythagorean Theorem: \(a^2 + b^2 = c^2\)
Substitute the known values:
\(a^2 + 24^2 = 40^2\)
\(a^2 + 576 = 1600\)
Subtract 576 from both sides to solve for \(a^2\):
\(a^2 = 1600 - 576\)
\(a^2 = 1024\)
To find 'a', take the square root of both sides:
\(a = \sqrt{1024}\)
\(a = 32\)
The width of the television screen is 32 inches.
Name Surname: .................................. Date: .... / .... / 202...
Pythagorean Theorem Worksheet
SCORE
A. True (T) / False (F)
( .... )
The Pythagorean Theorem applies to all types of triangles.
( .... )
The hypotenuse is always the longest side of a right-angled triangle.
( .... )
If the sides of a triangle are 3, 4, and 5 units, it is a right-angled triangle.
( .... )
In the Pythagorean Theorem formula \(a^2 + b^2 = c^2\), 'c' represents one of the legs.
( .... )
The Pythagorean Theorem can be used to find the length of a missing side if two sides of a right triangle are known.
B. Fill in the Blanks
1)
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the .................... of the squares of the lengths of the other two sides.
2)
The two shorter sides of a right-angled triangle that form the right angle are called .....................
3)
A triangle with side lengths 6, 8, and 10 units is a .................... triangle.
4)
The formula for the Pythagorean Theorem is \(a^2 + b^2 = ....................\).
5)
If a right triangle has legs of length 5 cm and 12 cm, its hypotenuse is .................... cm long.
C. Matching Concepts
( .... )
The longest side of a right-angled triangle, opposite the right angle.
- Pythagorean Triple
( .... )
The two shorter sides of a right-angled triangle that form the right angle.
- Hypotenuse
( .... )
An angle that measures exactly 90 degrees.
- Converse of the Pythagorean Theorem
( .... )
A set of three positive integers a, b, and c, such that \(a^2 + b^2 = c^2\).
- Legs
( .... )
If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.
- Right Angle
D. Short Answer Questions
1)
Explain when the Pythagorean Theorem can be applied to a triangle.
2)
What is a Pythagorean Triple? Provide an example.
E. Multiple Choice Questions
1)
A right-angled triangle has legs measuring 7 cm and 24 cm. What is the length of its hypotenuse?
A) 25 cmB) 31 cmC) 17 cmD) 20 cm
2)
Which of the following sets of side lengths could form a right-angled triangle?
A) 4, 5, 6B) 5, 12, 13C) 6, 8, 9D) 2, 3, 4
3)
The hypotenuse of a right triangle is 10 inches long, and one leg is 6 inches long. What is the length of the other leg?
A) 7 inchesB) 8 inchesC) 9 inchesD) 12 inches
F. Open-Ended Questions
1)
A ladder is leaning against a wall. The base of the ladder is 5 feet away from the wall, and the ladder reaches 12 feet up the wall. What is the length of the ladder?
2)
A rectangular park is 80 meters long and 60 meters wide. If a person walks diagonally across the park from one corner to the opposite corner, how far do they walk?
3)
A television screen has a diagonal measure of 40 inches. If the height of the screen is 24 inches, what is the width of the screen?