The perimeter of an isosceles right-angled triangle ...
The perimeter of an isosceles right-angled triangle is 2 meters. Find the length of its longer side.
A.
2 - √2
B.
-4 + 3√2
C.
It cannot be determined
D.
-2 + 2√2 m
Correct answer is D
Perimeter of a triangle = sum of all sides
⇒ P=y+x+x=2
⇒ y+2x=2
⇒ y=2−2x-----(i)
Using Pythagoras theorem
y2=x2+x2
⇒ y2=2x2
⇒ y=√2x2
⇒ y=x√2-----(ii)
Equate y
⇒ 2−2x=x√2
Square both sides
⇒ (2−2x)2=(x√2)2
⇒ 4−8x+4x2=2x2
⇒ 4−8x+4x2−2x2=0
⇒ 2x2−8x+4=0
⇒ x=−(−8)±√(−8)2−4×2×42×2
⇒ x=8±√324
⇒ x=8±4√24
⇒ x=2±√2
⇒ x=2+√2 or 2−√2
∴ x=2−√2 (for x has to be less than its perimeter)
∴ y=2−2x=2−2(2−√2)=−2+2√2
∴ The length of the longer side = -2 + 2√2m