Question

in the figure angle Q is equals to 90 degree angle P is equals to angle R is equals to 45 degree then PR is equals to​

Answer 1

Answer:

The length of PR is 5 √2 units.

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given that,

△PQR is a right-angled triangle.

m∟PQR = 90°

m∠QPR = m∠PRQ = 45°

PQ = 5 units

We have to find the length of PR.

Now, in △PQR,

m∟PQR = 90°

m∠QPR = m∠PRQ = 45°

∴ △PQR is 45-45-90° triangle.

By the property of 45-45-90° triangle,

The side opposite to the angle of measure 45° is ( 1 / √2 ) of the hypotenuse.

∴ PQ = 1 / √2 * PR

⇒ 5 = 1 / √2 * PR

⇒ PR = 5 * √2

⇒ PR = 5 √2 units

∴ The length of PR is 5 √2 units.

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Alternative Method:

In △PQR,

m∠QPR = m∠PRQ = 45° - - [ Given ]

∴ QR = PQ - - - [ Side opposite to congruent angles ]

⇒ QR = 5 units

Now, in △PQR, m∟PQR = 90°

∴ ( PR )² = ( PQ )² + ( QR )² - - [ Pythagoras theorem ]

⇒ ( PR )² = ( 5 )² + ( 5 )²

⇒ ( PR )² = 25 + 25

⇒ ( PR )² = 50

⇒ PR = √50 - - [ Taking square roots ]

⇒ PR = √( 25 × 2 )

⇒ PR = 5 √2 units

∴ The length of PR is 5 √2 units.