In the given figure, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is ______. - Mathematics

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In the given figure, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is ______.

Options

  • 6 cm

  • 9 cm

  • 4 cm

  • 2 cm

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Solution

In the given figure, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is 4 cm.

Explanation:


Given that, PR = 12 cm, QR = 6 cm and PL = 8 cm

Now, In right angled ΔPLR, using Pythagoras theorem,

(Hypotenuse)2 = (Perpendicular)2 + (Base)2

⇒ PR2 = PL2 + LR2

⇒ LR2 = PR2 – PL2 = (12)2 – (8)2

⇒ LR2 = 144 – 64 = 80

⇒ LR = `sqrt(80) = 4sqrt(5)` cm

∵ LR = LQ + QR

⇒ LQ = LR – QR = `(4sqrt(5) - 6)` cm

Now, area of ΔPLR,

A1 = `1/2 xx LR xx PL`

= `1/2 xx (4sqrt(5)) xx 8`

= `16sqrt(5)` cm2

Again, area of ΔPLQ,

A2 = `1/2 xx LQ xx PL`

= `1/2 xx (4sqrt(5) - 6) xx 8`

= `(16sqrt(5) - 24)` cm2

∴ Area of ΔPLR = Area of ΔPLQ + Area of ΔPQR

⇒ `16sqrt(5) = (16sqrt(5) - 24)` + Area of ΔPQR

⇒ Area of ΔPQR = 24 cm2

⇒ `1/2 xx PR xx QM` = 24

⇒ `1/2 xx 12 xx QM` = 24

∴  QM = 4 cm

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Chapter 9: Perimeter and Area - Exercise [Page 271]

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NCERT Exemplar Mathematics Class 7
Chapter 9 Perimeter and Area
Exercise | Q 16. | Page 271

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