Question

please answer me with solution​

Answer 1

Answer:

(a) Area of ∆PAR= 1/2*b*h=1/2*QR*PS

180=1/2*15*QR

QR=180*2/15=24cm

(b) Area of ∆PAR= 1/2*b*h=1/2*QR*PS

111=1/2*12*QR

QR=111*2/12=18.5cm

I hope my answer helps you ☺☺

Answer 2

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept}}}$

Here the concept or Areas of the Triangle has been used. We see that we are given two sub parts in the question where we have to find the length of a side. We are also given that there is one altitude and we have to find the other side. When we construct the triangle according to given notations, then we see that the side which we have to find that is QR is the base of triangle. Using this concept, we can get our answer.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{\pink{Area\;of\;Triangle\;=\;\bf{\dfrac{1}{2}\:\times\:Base\:\times\:Altitude}}}}$

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★ Solution :-

a.) ar(∆PQR) = 180 cm², PS = 15 cm

Given,

» Area of ∆PQR = 180 cm²

» Height of ∆PQR = Altitude = PS = 15 cm

From figure we see that,

• Base of the triangle ∆PQR = QR

We know that,

$\\\;\sf{:\rightarrow\;\;Area\;of\;Triangle\;=\;\bf{\dfrac{1}{2}\:\times\:Base\:\times\:Altitude}}$

By applying values, we get

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;\Delta\:PQR\;=\;\bf{\dfrac{1}{2}\:\times\:QR\:\times\:PS}}$

$\\\;\sf{:\Longrightarrow\;\;180\;=\;\bf{\dfrac{1}{2}\:\times\:QR\:\times\:15}}$

By cross multiplication, we get

$\\\;\sf{:\Longrightarrow\;\;QR\;=\;\bf{\dfrac{180\:\times\:2}{15}}}$

$\\\;\sf{:\Longrightarrow\;\;QR\;=\;\bf{12\:\times\:2}}$

$\\\;\sf{:\Longrightarrow\;\;QR\;=\;\bf{\red{24\;\:cm}}}$

$\\\;\boxed{\underline{\tt{Hence,\;\:length\;\:of\;\:QR\;=\;\bf{\purple{24\;\:cm}}}}}$

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b.) ar(∆PQR) = 111 cm², PS = 12 cm

Given,

» Area of ∆PQR = 111 cm²

» Height of ∆PQR = Altitude = PS = 12 cm

From figure we see that,

• Base of ∆PQR = QR

We know that,

$\\\;\sf{:\rightarrow\;\;Area\;of\;Triangle\;=\;\bf{\dfrac{1}{2}\:\times\:Base\:\times\:Altitude}}$

By applying values, we get

$\\\;\sf{:\Longrightarrow\;\;Area\;of\;\Delta\:PQR\;=\;\bf{\dfrac{1}{2}\:\times\:QR\:\times\:PS}}$

$\\\;\sf{:\Longrightarrow\;\;111\;=\;\bf{\dfrac{1}{2}\:\times\:QR\:\times\:12}}$

$\\\;\sf{:\Longrightarrow\;\;111\;=\;\bf{QR\:\times\:6}}$

By cross multiplication, we get

$\\\;\sf{:\Longrightarrow\;\;QR\;=\;\bf{\dfrac{111}{6}}}$

$\\\;\sf{:\Longrightarrow\;\;QR\;=\;\bf{\green{18.5\;\:cm}}}$

$\\\;\boxed{\underline{\tt{Hence,\;\:length\;\:of\;\:QR\;=\;\bf{\orange{18.5\;\:cm}}}}}$

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★ More to know :-

• Formulas ::

$\\\;\tt{\leadsto\;\;Perimeter\;of\;\Delta\;=\;Sum\;of\;Sides}$

$\\\;\tt{\leadsto\;\;Area\;of\;Equilateral\;\Delta\;=\;a^{2}\:\dfrac{\sqrt{3}}{4}}$

$\\\;\tt{\leadsto\;\;Semi\:-\:Perimeter\;of\;\Delta\;=\;\dfrac{Perimeter}{2}}$

$\\\;\tt{\leadsto\;\;Area\;of\;\Delta\;=\;\sqrt{s(s\:-\:a)(s\:-\:b)(s\:-\:c)}}$

• Properties of Triangle ::

• Its is made up of three sides.

• A triangle whose all three sides are equal is known as Equilateral Triangle.

• A triangle whose two sides are equal is known as Isoceles Triangle.

• A triangle whose all three sides are of different measures, is known as Scalene Triangle.