Question

Use the Pythagoras Theorem to check which of the following triplets would make a right triangle.
A) \[5,20,25\]
B) \[7,24,25\]
C) \[7,23,25\]
D) \[15,20,25\]

Answer 1

Hint: In order to find the triplet we need to use the Pythagoras theorem. For example in three numbers, the square of one number is equal to the sum of squares of the other two numbers then these three numbers represent a right angle triplet or Pythagoras triplet. This is also the rule of Pythagoras Theorem. In a right angle triangle the square of the bigger side is equal to the sum of squares of the other two sides. The bigger side is known as hypotenuse, horizontal side is base and last one side is known as perpendicular.

Complete step-by-step answer:
Pythagoras Theorem:
In a right angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the triangle.
 In triangle \[ABC\]
\[{{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(Perpendicular)}^{2}}\]
\[{{(AC)}^{2}}={{(BC)}^{2}}+{{(AB)}^{2}}\]
A.\[5,20,25\]
Let hypotenuse \[=25\]
\[Base=5\]
Perpendicular \[=20\]
Use the Pythagoras theorem
\[{{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(perpendicular)}^{2}}\]
Substitute the values of hypotenuse, base and perpendicular in the given equation
\[\Rightarrow {{(25)}^{2}}={{(5)}^{2}}+{{(20)}^{2}}\]
Rewrite the expression after open the squares
\[\Rightarrow 625=25+400\]
Rewrite the expression after simplification
\[\Rightarrow 625=425\]
This is not possible so this is not the correct answer.
B. \[7,24,25\]
Let hypotenuse \[=25\]
Base \[=7\]
Perpendicular \[=24\]
Using the Pythagoras Theorem
\[{{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(perpendicular)}^{2}}\]
\[\Rightarrow {{(25)}^{2}}={{(7)}^{2}}+{{(24)}^{2}}\]
Rewrite the expression after open the squares
\[\Rightarrow 625=49+576\]
Rewrite the expression after simplification
\[\Rightarrow 625=625\]
This is true so this is the correct answer.
C.\[7,23,25\]
Let hypotenuse \[=25\]
Base \[=7\]
Perpendicular \[=23\]
Using the Pythagoras Theorem
\[{{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(perpendicular)}^{2}}\]
\[\Rightarrow {{(25)}^{2}}={{(7)}^{2}}+{{(23)}^{2}}\]
Rewrite the expression after open the squares
\[\Rightarrow 625=49+529\]
Rewrite the expression after simplification
\[\Rightarrow 625=578\]
This is not possible so this is not the correct answer.
D. \[15,20,25\]
Let hypotenuse \[=25\]
Base \[=15\]
Perpendicular \[=20\]
Using the Pythagoras Theorem
\[{{(Hypotenuse)}^{2}}={{(Base)}^{2}}+{{(perpendicular)}^{2}}\]
\[\Rightarrow {{(25)}^{2}}={{(15)}^{2}}+{{(20)}^{2}}\]
Rewrite the expression after open the squares
\[\Rightarrow 625=225+400\]
Rewrite the expression after simplification
\[\Rightarrow 625=625\]
This is true so this is the correct answer.

Hence, Options B and D are the correct answer.

Note: This type of problem is also solved with the help of the trigonometric formula. With the use of trigonometric ratios we can find the right angle triangle triplet. In the Pythagoras triplet we check the square of one number is equal to the sum of the squares of the other two numbers that is also called a right angle triplet.
Below are a few pythagorean triples- \[(3,4,5)\] \[(6,8,10)\] \[(5,12,13)\]
\[(20,21,29)\]