Solve for RT in triangle RET where EY is a median, RY = 2z - 1, and TY = 4z - 11.
Understand the Problem
The question is asking to solve for the length of RT in triangle RET, given that EY is a median and provided specific lengths for segments RY and TY in terms of the variable z.
Answer
The length of \( RT \) is \( 18 \).
Answer for screen readers
The length of ( RT ) is ( 18 ).
Steps to Solve
- Understanding the median in a triangle
In triangle RET, EY is a median, which means that EY divides segment RT into two equal parts. Thus, we can express this relationship as: $$ RY = TY $$
- Setting up the equation
From the problem, we have:
- $RY = 2z - 1$
- $TY = 4z - 11$
Since ( RY = TY ), we can set up the equation: $$ 2z - 1 = 4z - 11 $$
- Solving for z
Now, we solve the equation for ( z ).
Subtract ( 2z ) from both sides: $$ -1 = 2z - 11 $$
Add ( 11 ) to both sides: $$ 10 = 2z $$
Now, divide both sides by 2: $$ z = 5 $$
- Finding the lengths of RY and TY
Now that we have ( z = 5 ), we can substitute back to find ( RY ) and ( TY ):
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For ( RY ): $$ RY = 2(5) - 1 = 10 - 1 = 9 $$
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For ( TY ): $$ TY = 4(5) - 11 = 20 - 11 = 9 $$
- Finding the length of RT
Since ( RT = RY + TY ): $$ RT = 9 + 9 = 18 $$
The length of ( RT ) is ( 18 ).
More Information
The median of a triangle has the property that it divides the opposite side into two equal lengths. This property is useful in calculations involving side lengths when one side is expressed as a variable.
Tips
- Confusing the expressions for RY and TY can lead to incorrect equations. Ensure you correctly set ( RY ) equal to ( TY ).
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