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
Answers
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.
