Question

ps is a midsegment of \\( \triangle qrt \\).
if \\( qr=-2z + 64 \\) and \\( ps=z - 26 \\), what is the value of \\( z \\)?

Explanation:

Step1: Apply the midsegment theorem

The midsegment theorem states that the length of a midsegment of a triangle is half the length of the parallel side. So, \( QR = 2\times PS \).

Step2: Substitute the given expressions

Substitute \( QR=-2z + 64 \) and \( PS=z - 26 \) into \( QR = 2\times PS \). We get \( -2z+64=2(z - 26) \).

Step3: Expand the right - hand side

Using the distributive property \( a(b + c)=ab+ac \), \( 2(z - 26)=2z-52 \). So the equation becomes \( -2z + 64=2z-52 \).

Step4: Add \( 2z \) to both sides

\( -2z+2z + 64=2z+2z-52 \), which simplifies to \( 64 = 4z-52 \).

Step5: Add 52 to both sides

\( 64 + 52=4z-52 + 52 \). So \( 116=4z \).

Step6: Solve for \( z \)

Divide both sides by 4. \( z=\frac{116}{4}=29 \).

Answer:

\( 29 \)