The value of y is: 4 units
What is the length of the triangle?
The given information:
NR is perpendicular to MN
NS = 6 units
SM = 2 units
RM = x units
NR = y units
RS = z units
Use the Pythagorean theorem on right triangle NRS:
y² = RS² + NS²
y² = z² + 6²
Use the Pythagorean theorem on right triangle RMS:
x² = RS²+ SM²
x² = z² + 2² ___________(1)
Since NR is the hypotenuse of both right triangles NRS and RMS, we can set the two expressions for z² equal to each other:
y² = x² ___________(2)
Substitute the expression for x² in equation 1:
y² = z² + 2²
Solve for y:
y = √(z² + 2²)
Since MRN is half of an equilateral triangle, we know that MN = 2x:
MN = 2x
Use the Pythagorean theorem on right triangle MNR:
y² = RM² + (MN/2)²
y² = x² + (x)²
y² = 2x²
Set the two expressions for y² equal to each other:
y² = y²
z² + 2² = 2x²
Substitute the expression for x² from equation 1:
z² + 2² = 2(z²+ 2²)
Simplify and solve for z²:
z² + 2² = 2z² + 4²
-z² = 4² - 2²
z² = 12
Solve for z:
z = √12
z = 2√3
Substitute the value of z back into the equation
y = √(z² + 2²)
y = √(2√3)² + 2²
y = √(12 + 4)
y = √16
y = 4
From equation 2 we can say that
y = x
Hence,
y = x = 4 units
Option B is the right choice.