Triangle MRN is created when an equilateral triangle is folded in half.
What is the value of y?
A. 2√3 units
B. 4 units
C. 4√3 units
D. 8 units
(Image: Triangle N R M is shown. Angle N R M is a right angle. An altitude is drawn from point R to point S on side M N to form a right angle. The length of N S is 6, the length of S M is 2, the length of R M is x, the length of N R is y, and the length of R S is z.)
Triangle MRN is created when an equilateral triangle is folded in half. What is the value of y?
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To find the value of ( y ) in triangle MRN, we can use the properties of right triangles and the information given.
1. Since triangle MRN is formed from folding an equilateral triangle in half, it is a right triangle with ( angle NRM ) as a right angle.
2. We know from the problem:
– ( NS = 6 ) (length from point N to S)
– ( SM = 2 ) (length from point S to M)
3. The full length of side ( MN ) is ( NS + SM = 6 + 2 = 8 ).
4. In right triangle ( RNS ), by using the Pythagorean theorem:
[
NR^2 + RS^2 = NS^2
]
Plugging the values in, we have:
[
y^2 + z^2 = 6^2
]
[
y^2 + z^2 = 36 , text{(1)}
]
5. Since we have ( SM = 2 ), we can also express ( RM ) in terms of ( y ):
[
RM^2 = y^2 + 2^2
]
[
RM^2 = y^2 + 4 , text{(2)}
]
6. From