How to Apply the Pythagorean Theorem in Real-World Math Problems
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: Step-by-Step Practice
1. What is the Pythagorean theorem in simple terms for kids?
The Pythagorean theorem is a special rule in Maths that applies only to right-angled triangles. It states that if you square the lengths of the two shorter sides (the 'legs') and add them together, the total will be equal to the square of the length of the longest side (the 'hypotenuse'). The formula is famously known as a² + b² = c².
2. What are the 'legs' and the 'hypotenuse' of a right-angled triangle?
In a right-angled triangle, the two sides that meet to form the 90-degree angle are called the legs (often labelled 'a' and 'b'). The side opposite the right angle, which is always the longest side of the triangle, is called the hypotenuse (labelled 'c'). Correctly identifying the hypotenuse is the most important step in using the theorem.
3. How do you use the Pythagorean theorem to find a missing side?
To find a missing side using the Pythagorean theorem, follow these simple steps:
- Step 1: Confirm you have a right-angled triangle.
- Step 2: Identify which sides are the legs (a, b) and which is the hypotenuse (c).
- Step 3: Write down the formula: a² + b² = c².
- Step 4: Substitute the lengths of the two known sides into the formula.
- Step 5: Solve the equation to find the length of the missing side. Remember to calculate the square root at the end if you are finding 'c'.
4. What is a simple real-world example of the Pythagorean theorem?
Imagine you are leaning a ladder against a wall. The wall is straight, and the ground is flat, forming a right angle. If the base of the ladder is 3 feet from the wall (side 'a') and the ladder reaches 4 feet up the wall (side 'b'), you can find the ladder's length (the hypotenuse 'c'). Using the theorem: 3² + 4² = 9 + 16 = 25. The square root of 25 is 5. Therefore, the ladder is 5 feet long.
5. Why does the Pythagorean theorem only work for right-angled triangles?
The Pythagorean theorem is based on the unique geometric properties of a 90-degree angle. The specific relationship where the sum of the squares of the two shorter sides equals the square of the longest side only holds true when those two sides are perpendicular. For triangles with other angles (acute or obtuse), the sides have a different relationship, and the formula a² + b² = c² will not give a correct result.
6. What is the most common mistake students make when using the Pythagorean theorem?
The most common mistake is incorrectly identifying the hypotenuse. Students often substitute one of the leg lengths for 'c' in the formula a² + b² = c². It is crucial to remember that 'c' must always be the side opposite the right angle. Another frequent error is performing all the calculations correctly but forgetting to take the final square root to find the length of the side, leaving the answer as c² instead of c.
7. Besides finding lengths, what is the importance of learning the Pythagorean theorem?
Learning the Pythagorean theorem is important because it forms the foundation for more advanced topics in geometry, trigonometry, and even physics. It helps develop logical reasoning and problem-solving skills. Understanding this theorem is the first step to understanding concepts like distances in a coordinate plane, vector calculations, and spatial reasoning, which are useful in fields like architecture, engineering, and computer graphics.
