How to Use the Pythagorean Theorem to Find a Missing Leg in Right Triangles
According to the Pythagoras theorem, the square of the hypotenuse equals the sum of the squares of the remaining two sides in a right-angled triangle.
Let's solve some questions for further understanding -
Questions:
Find the value of y in the given triangle.
Image - an image of a triangle
Identify the length of the hypotenuse?
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What is the value of the missing side of the triangle?
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Find the value of x.
Image - an image of a triangle
Find the value of c.
Image - an image of a triangle
Find the value of the missing side.
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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.
Use Pythagoras Theorem to find the missing side.
Image - an image of a triangle
Find the third side of the given triangle.
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Is ∆ABC a right angled triangle, where AB = 5cm, BC = 10cm, AC = 15cm?
What is the length of the hypotenuse in the given triangle?
Image - an image of a triangle
Find the missing side by using Pythagoras Theorem.
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What is the length of the hypotenuse of the given triangle ?
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ABC is a right triangle. AC is its hypotenuse. Length of side AB is 2√5. Side BC is twice of side AB. Find the length of AC.
Find the length of the hypotenuse.
Image - an image of a triangle
Answers -
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.
5.1mm
By Pythagoras Theorem, a2+b2=c2
Let a=3 b= 6
9 + 36 = c2 ⇒ 45 = c2
√45 = c
By the Pythagoras Theorem, Hypotenuse2 = Base2 + Perpendicular2
x2 = 64+36 = 100
x = 10
Therefore, the value of x is 10.
By Pythagoras Theorem, a2+b2=c2
Let a=5 b=7
25 + 49 = c2 ⇒ 74 = c2
√74 = c
Therefore, the value of c is √74.
By Pythagoras Theorem, a2+b2=c2
Let a=5 b=5
25 + 25 =c2 ⇒ 50 = c2
√50 = c
4cm.
By Pythagoras Theorem, a2+b2=c2
Let c =19 b=17
a2 + 289 = 361 ⇒ 72 = a2
√72 = a
Given: a = 3, c = 5
Putting the values in the Pythagoras Formula: a2+b2=c2
9 + b2 = 25
b2 = 25 − 9 = 16
b = 4
Therefore, the third side is 4.
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.
9.6m.
By Pythagoras Theorem, a2+b2=c2
Let c =20 b=10
a2 + 100 = 400 ⇒ 300 = a2
√300 = a
Pythagoras Formula: a2+b2=c2
Let AB = a = 20, BC = b = 21
Putting values in the formula
400 + 441 = c2
841 = c2
c = 29 cm
AC = 10
10in
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The experts at Vedantu have solved all the questions to provide easy reference to the students. KG-3 students can refer to these solutions to clarify any doubts that they might have about the chapter.
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FAQs on Pythagorean Theorem Find the Missing Leg Worksheet Practice
1. What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right triangle, a² + b² = c², where c is the hypotenuse. In this formula:
- a and b are the legs (shorter sides).
- c is the hypotenuse (longest side opposite the right angle).
2. How do you find the missing leg using the Pythagorean Theorem?
To find a missing leg, use the formula a = √(c² − b²) or b = √(c² − a²). Follow these steps:
- Step 1: Square the hypotenuse.
- Step 2: Subtract the square of the known leg.
- Step 3: Take the square root of the result.
3. What is the formula for finding a missing leg?
The formula for a missing leg in a right triangle is leg = √(hypotenuse² − other leg²). In symbols:
- a = √(c² − b²)
- b = √(c² − a²)
4. Can you give an example of finding a missing leg?
Yes, to find a missing leg, substitute known values into a² + b² = c² and solve. Example:
- Given: c = 13, b = 5
- Step 1: 13² = 169
- Step 2: 5² = 25
- Step 3: a² = 169 − 25 = 144
- Step 4: a = √144 = 12
5. How do you know which side is the hypotenuse?
The hypotenuse is the longest side of a right triangle and is always opposite the right angle. To identify it:
- Look for the 90° angle.
- The side directly across from that angle is the hypotenuse.
6. Can you use the Pythagorean Theorem on any triangle?
No, the Pythagorean Theorem only works for right triangles. A right triangle must have one 90° angle. If the triangle does not contain a right angle, this formula cannot be applied directly.
7. What are common mistakes when finding a missing leg?
Common mistakes when solving for a missing leg include subtracting incorrectly or mixing up the hypotenuse. Watch out for:
- Using a² + b² = c² incorrectly when solving for a leg.
- Forgetting to subtract instead of add.
- Not taking the square root at the final step.
- Choosing the wrong side as the hypotenuse.
8. Why do you subtract when finding a missing leg?
You subtract because you are rearranging a² + b² = c² to isolate one leg. If solving for a, then:
- a² = c² − b²
9. How do you check your answer when finding a missing leg?
To check your answer, substitute all three side lengths back into a² + b² = c². Example:
- If sides are 5, 12, and 13
- 5² + 12² = 25 + 144 = 169
- 13² = 169
10. Where is the Pythagorean Theorem used in real life?
The Pythagorean Theorem is used in construction, navigation, engineering, and distance calculations. Real-life applications include:
- Finding the length of a ladder against a wall.
- Calculating diagonal distances on maps.
- Determining shortest paths in coordinate geometry.
