Pythagorean Theorem formula steps and solved worksheet examples
The Pythagoras Theorem applied in a right-angled triangle states that the square of the side hypotenuse will be equal to the sum of the squares of the base and the perpendicular.
Let's solve some questions for further understanding -
Questions:
Identify the length of hypotenuse in the given triangle -
Image - An Image of a Triangle
Pythagoras' theorem is applicable in which kind of triangles?
Solve for an in the given triangle -
Image - An Image of a Triangle
Is the given figure a right-angled triangle ?
Image - An Image of a Triangle
Find the value of the missing side.
Image - An Image of a Triangle
Find the length of the hypotenuse.
Image - An Image of a Triangle
Use Pythagorean Theorem to check whether the given triangle is a right-angle triangle or not.
Image - An Image of a Triangle
Is the following Pythagorean Triplet correct?
3, 5 and 7
Solve for c in the given triangle -
Image - An Image of a Triangle
In the below given right triangle, find the value of y.
Image - An Image of a Triangle
In a right triangle, the longest side is 8 cm. One of the remaining sides is 4√3 cm long. Find the length of the other side.
What is the value of the missing side of the triangle?
Image - An Image of a Triangle
ABC is a right triangle. AC is its hypotenuse. The length of side AB is 2√5. Side BC is twice side AB. Find the length of AC.
Find the value of x.
Image - An Image of a Triangle
Is ∆ABC a right-angled triangle, where AB = 5cm, BC = 10cm, AC = 15cm?
Answers -
7.2, the length of the hypotenuse in the given triangle is 7.2.
Pythagoras theorem is applicable only in right-angled triangles.
Given, b=12 and c= 15
a2+b2=c2
a2+ 144 =225
Subtract 144 from each side to get:
144 - 144 + a2 = 225 - 144
a2 = 225 - 144 ⇒ a2 = 81
⇒ a = √81
a = 9
Yes, the given triangle is a right angled triangle.
By Pythagoras theorem, a2+b2=c2
Let b=9 c=12
144 - 81 =a2 ⇒ 63 = a2
√63 = a
Therefore, the length of the missing side is √63.
10in.
By Pythagoras theorem, a2+b2=c2
Let a=6 b=7 c=9
36 + 49 = 81 ⇔ 9 x 9 = 81
Therefore, LHS≠RHS
Hence, the given triangle is not a right angle triangle.
By Pythagoras theorem, a2+b2=c2
Let a=3 b=5 c=7
9 + 25 = 34
7 × 7 = 49
Therefore, LHS≠RHS
Hence, 3, 5 and 7 are not correct Pythagorean triplets.
Given, a=11 b=15
By Pythagoras theorem,
a2+b2=c2
121 + 225 = c2
346 = c2
c = √346m
By Pythagoras theorem we get,
z2=x2+y2
Now, directly substituting the values
⇒ 169 = 25 + y2
⇒ y2 = 144 ⇒ y = 12
Hence, the value of y is 12.
4cm.
By Pythagoras theorem, a2+b2=c2
Let a=3 b= 6
9 + 36 = c2 ⇒ 45 = c2
√45 = c
Therefore, the length of the missing side is √45cm.
AC = 10
By the pythagoras theorem, Hypotenuse2 = Base2 + Perpendicular2
x2 = 64+36 = 100
x = 10
Therefore, the value of x is 10.
If ∆ABC is a right-angled triangle, then AB2+BC2=AC2
AB = 5, BC =10, AC = 15
AB2 + BC2 = 25 + 100 = 125
AC2 = 225
Since, AC2 ≠ AB2 + BC2
Hence, ABC is not a right-angled triangle.
Importance of Pythagoras Theorem Worksheet for Students
Students of KG-3 will be able to gain a better understanding of the Pythagoras Theorem from these worksheets. From these questions, students will learn how to find the different sides of the right-angled triangle. The Pythagoras theorem topics have been explained in the worksheets to understand the students better. Not just that, but there are different types of sums, such as word problems, real-time applications, numerical questions, and so much more. Students will also find MCQ on Pythagoras' Theorem in these worksheets.
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Benefits of Pythagoras Theorem Questions and Worksheets
For students who are in Class KG-3, it is extremely important to complete their syllabus. Thus they can take some help from the questions provided in the worksheets related to the Pythagoras theorem and understand the chapter completely.
These worksheets have been prepared by the learned subject matter experts at Vedantu and contain all the details about the chapter. The experts have also included some special tricks and shortcuts to solve the problems easily.
Students can download and refer to the worksheets to resolve any queries or clarify any doubts that they might have about the chapter. This will enable them to boost their preparation for the examination best.
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Download Worksheets for Pythagoras Theorem Questions
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FAQs on Pythagorean Theorem Worksheet Practice and Solutions
1. What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, written as a² + b² = c².
- It applies only to right-angled triangles.
- a and b are the legs (shorter sides).
- c is the hypotenuse (longest side).
- This formula is used to find a missing side length.
2. What is the formula for the Pythagorean Theorem?
The formula for the Pythagorean Theorem is a² + b² = c².
- a and b represent the perpendicular sides.
- c represents the hypotenuse.
- To find a missing side, rearrange the formula:
- c = √(a² + b²)
- a = √(c² − b²)
3. How do you use the Pythagorean Theorem to find the hypotenuse?
To find the hypotenuse, substitute the leg lengths into c = √(a² + b²).
- Example: If a = 3 and b = 4, then:
- c = √(3² + 4²)
- c = √(9 + 16)
- c = √25
- c = 5
4. How do you find a missing leg using the Pythagorean Theorem?
To find a missing leg, use the formula a = √(c² − b²) or b = √(c² − a²).
- Example: If c = 10 and b = 6:
- a = √(10² − 6²)
- a = √(100 − 36)
- a = √64
- a = 8
5. What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c².
- Common examples include:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
6. Does the Pythagorean Theorem only work for right triangles?
Yes, the Pythagorean Theorem works only for right-angled triangles.
- One angle must measure exactly 90°.
- If the triangle is not right-angled, the formula a² + b² = c² does not apply.
- For non-right triangles, use the Law of Cosines instead.
7. Can the Pythagorean Theorem be used to check if a triangle is right-angled?
Yes, a triangle is right-angled if its sides satisfy a² + b² = c².
- Identify the longest side as c.
- Square all three sides.
- If the sum of the squares of the shorter sides equals the square of the longest side, the triangle is right-angled.
- Example: 6, 8, 10 → 6² + 8² = 36 + 64 = 100 = 10² ✔
8. What are some real-life applications of the Pythagorean Theorem?
The Pythagorean Theorem is used to calculate distances in real-life right triangle situations.
- Finding the length of a ladder against a wall.
- Calculating the diagonal of a rectangle.
- Determining distance between two points on a coordinate plane.
- Construction, engineering, and architecture measurements.
9. How is the Pythagorean Theorem used on the coordinate plane?
On the coordinate plane, the Pythagorean Theorem is used to find the distance between two points using the distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
- This formula comes directly from a² + b² = c².
- Example: Distance between (1,2) and (4,6):
- d = √[(4−1)² + (6−2)²]
- d = √(9 + 16)
- d = 5
10. What are common mistakes when using the Pythagorean Theorem?
Common mistakes include misidentifying the hypotenuse or forgetting to square and square root correctly.
- The hypotenuse is always the longest side.
- Do not add side lengths before squaring.
- Remember to take the square root at the end when solving for a side.
- Ensure the triangle has a 90° angle before applying the formula.
