Answer
Verified
417.9k+ views
Hint: Here, we will proceed by firstly finding all the sides of the triangle PQR using the concept that the perimeter of any triangle is equal to the sum of all the sides and then we will use the formula for the area of any triangle i.e., Area of the triangle = $\dfrac{1}{2} \times $(Base)$ \times $(Height).
Complete step-by-step answer:
Let us suppose a right triangle PQR which is right angled at vertex Q.
Given, Perimeter of the right triangle PQR = 60 cm
Hypotenuse of the right triangle PQR, PR = 25 cm
Since, the perimeter of any triangle is equal to the sum of all the three sides of the triangle.
i.e., Perimeter of the right triangle PQR = PQ + QR + PR
$ \Rightarrow $60 = PQ + QR + 25
$ \Rightarrow $PQ +QR = 60 -25
$ \Rightarrow $PQ +QR = 35
$ \Rightarrow $PQ = (35 – QR) $ \to (1)$
By Pythagoras Theorem in any right triangle, we have
${\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Perpendicular}}} \right)^2} + {\left( {{\text{Base}}} \right)^2}$
Using the Pythagoras Theorem in right triangle PQR, we get
\[{\left( {{\text{PR}}} \right)^2} = {\left( {{\text{PQ}}} \right)^2} + {\left( {{\text{QR}}} \right)^2}\]
By putting PR = 25 and substituting equation (1) in the above equation, we get
\[
\Rightarrow {\left( {{\text{25}}} \right)^2} = {\left( {{\text{35}} - {\text{QR}}} \right)^2} + {\left( {{\text{QR}}} \right)^2} \\
\Rightarrow 625 = {\left( {{\text{35}}} \right)^2} + {\left( {{\text{QR}}} \right)^2} - 2\left( {35} \right)\left( {{\text{QR}}} \right) + {\left( {{\text{QR}}} \right)^2} \\
\Rightarrow 2{\left( {{\text{QR}}} \right)^2} - 70\left( {{\text{QR}}} \right) - 625 + 1225 = 0 \\
\Rightarrow 2{\left( {{\text{QR}}} \right)^2} - 70\left( {{\text{QR}}} \right) + 600 = 0 \\
\Rightarrow {\left( {{\text{QR}}} \right)^2} - 35\left( {{\text{QR}}} \right) + 300 = 0 \\
\Rightarrow {\left( {{\text{QR}}} \right)^2} - 15\left( {{\text{QR}}} \right) - 20\left( {{\text{QR}}} \right) + 300 = 0 \\
\Rightarrow \left( {{\text{QR}}} \right)\left( {{\text{QR}} - 15} \right) - 20\left( {{\text{QR}} - 15} \right) = 0 \\
\Rightarrow \left( {{\text{QR}} - 15} \right)\left( {{\text{QR}} - 20} \right) = 0 \\
\]
Either \[
{\text{QR}} - 15 = 0 \\
\Rightarrow {\text{QR}} = 15{\text{ cm}} \\
\] or \[
{\text{QR}} - 20 = 0 \\
\Rightarrow {\text{QR}} = 20{\text{ cm}} \\
\]
Put QR = 15 cm in equation (1), we get
$ \Rightarrow $PQ = (35 – 15) = 20 cm
Put QR = 20 cm in equation (1), we get
$ \Rightarrow $PQ = (35 – 20) = 15 cm
As, we know that the area of any triangle is given by
Area of the triangle = $\dfrac{1}{2} \times $(Base)$ \times $(Height)
For right angled triangle PQR with PQ = 20 cm, QR = 15 cm and PR = 25 cm,
Base = QR = 15 cm, Height = PQ = 20 cm
So, Area of the triangle PQR = $\dfrac{1}{2} \times $(QR)$ \times $(PQ) = $\dfrac{1}{2} \times $(15)$ \times $(20) = 150 ${\text{c}}{{\text{m}}^2}$
For right angled triangle PQR with PQ = 15 cm, QR = 20 cm and PR = 25 cm,
Base = QR = 15 cm, Height = PQ = 20 cm
So, Area of the triangle PQR = $\dfrac{1}{2} \times $(QR)$ \times $(PQ) = $\dfrac{1}{2} \times $(20)$ \times $(15) = 150 ${\text{c}}{{\text{m}}^2}$
In both the cases, the value of the area of the triangle PQR obtained is the same which is equal to 150 ${\text{c}}{{\text{m}}^2}$.
Note: In any right angled triangle, the side opposite to the right angle is known as the hypotenuse, the side opposite to the considered acute angle is known as the perpendicular and the remaining side is known as the base. Here, in this case the hypotenuse is the side PR, the perpendicular and the base of this triangle will depend on the considered acute angle.
Complete step-by-step answer:
Let us suppose a right triangle PQR which is right angled at vertex Q.
Given, Perimeter of the right triangle PQR = 60 cm
Hypotenuse of the right triangle PQR, PR = 25 cm
Since, the perimeter of any triangle is equal to the sum of all the three sides of the triangle.
i.e., Perimeter of the right triangle PQR = PQ + QR + PR
$ \Rightarrow $60 = PQ + QR + 25
$ \Rightarrow $PQ +QR = 60 -25
$ \Rightarrow $PQ +QR = 35
$ \Rightarrow $PQ = (35 – QR) $ \to (1)$
By Pythagoras Theorem in any right triangle, we have
${\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Perpendicular}}} \right)^2} + {\left( {{\text{Base}}} \right)^2}$
Using the Pythagoras Theorem in right triangle PQR, we get
\[{\left( {{\text{PR}}} \right)^2} = {\left( {{\text{PQ}}} \right)^2} + {\left( {{\text{QR}}} \right)^2}\]
By putting PR = 25 and substituting equation (1) in the above equation, we get
\[
\Rightarrow {\left( {{\text{25}}} \right)^2} = {\left( {{\text{35}} - {\text{QR}}} \right)^2} + {\left( {{\text{QR}}} \right)^2} \\
\Rightarrow 625 = {\left( {{\text{35}}} \right)^2} + {\left( {{\text{QR}}} \right)^2} - 2\left( {35} \right)\left( {{\text{QR}}} \right) + {\left( {{\text{QR}}} \right)^2} \\
\Rightarrow 2{\left( {{\text{QR}}} \right)^2} - 70\left( {{\text{QR}}} \right) - 625 + 1225 = 0 \\
\Rightarrow 2{\left( {{\text{QR}}} \right)^2} - 70\left( {{\text{QR}}} \right) + 600 = 0 \\
\Rightarrow {\left( {{\text{QR}}} \right)^2} - 35\left( {{\text{QR}}} \right) + 300 = 0 \\
\Rightarrow {\left( {{\text{QR}}} \right)^2} - 15\left( {{\text{QR}}} \right) - 20\left( {{\text{QR}}} \right) + 300 = 0 \\
\Rightarrow \left( {{\text{QR}}} \right)\left( {{\text{QR}} - 15} \right) - 20\left( {{\text{QR}} - 15} \right) = 0 \\
\Rightarrow \left( {{\text{QR}} - 15} \right)\left( {{\text{QR}} - 20} \right) = 0 \\
\]
Either \[
{\text{QR}} - 15 = 0 \\
\Rightarrow {\text{QR}} = 15{\text{ cm}} \\
\] or \[
{\text{QR}} - 20 = 0 \\
\Rightarrow {\text{QR}} = 20{\text{ cm}} \\
\]
Put QR = 15 cm in equation (1), we get
$ \Rightarrow $PQ = (35 – 15) = 20 cm
Put QR = 20 cm in equation (1), we get
$ \Rightarrow $PQ = (35 – 20) = 15 cm
As, we know that the area of any triangle is given by
Area of the triangle = $\dfrac{1}{2} \times $(Base)$ \times $(Height)
For right angled triangle PQR with PQ = 20 cm, QR = 15 cm and PR = 25 cm,
Base = QR = 15 cm, Height = PQ = 20 cm
So, Area of the triangle PQR = $\dfrac{1}{2} \times $(QR)$ \times $(PQ) = $\dfrac{1}{2} \times $(15)$ \times $(20) = 150 ${\text{c}}{{\text{m}}^2}$
For right angled triangle PQR with PQ = 15 cm, QR = 20 cm and PR = 25 cm,
Base = QR = 15 cm, Height = PQ = 20 cm
So, Area of the triangle PQR = $\dfrac{1}{2} \times $(QR)$ \times $(PQ) = $\dfrac{1}{2} \times $(20)$ \times $(15) = 150 ${\text{c}}{{\text{m}}^2}$
In both the cases, the value of the area of the triangle PQR obtained is the same which is equal to 150 ${\text{c}}{{\text{m}}^2}$.
Note: In any right angled triangle, the side opposite to the right angle is known as the hypotenuse, the side opposite to the considered acute angle is known as the perpendicular and the remaining side is known as the base. Here, in this case the hypotenuse is the side PR, the perpendicular and the base of this triangle will depend on the considered acute angle.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Select the word that is correctly spelled a Twelveth class 10 english CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What organs are located on the left side of your body class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE