Question

1. find the indicated length. find st. r 4x s 32 12x t 26. find ac. a 14 4x + 4 b 3x - 4 c 27. find np. m x - 5 n 6x - 23 3x + 2 p point j is between h and k on hk. use the given information to write an

Explanation:

Step1: Analyze the first - line segment problem (25)

Given that \(RS + ST=RT\), where \(RS = 4x\), \(ST\) is unknown, and \(RT = 32\) and \(RS=12x - 32\). First, set up the equation based on the segment - addition postulate: \(4x+ST = 32\). Also, since \(4x=12x - 32\), we solve for \(x\).
\[

$$\begin{align*} 12x-4x&=32\\ 8x&=32\\ x& = 4 \end{align*}$$

\]

Step2: Find \(ST\)

Substitute \(x = 4\) into \(4x+ST = 32\). Since \(4x=4\times4 = 16\), then \(ST=32 - 4x\). So \(ST=32-16=16\).

Step3: Analyze the second - line segment problem (26)

Given \(AB + BC=AC\), where \(AB = 14\), \(BC = 3x - 4\), and \(AC=4x + 4\). Set up the equation: \(14+(3x - 4)=4x + 4\).
\[

$$\begin{align*} 14+3x-4&=4x + 4\\ 3x + 10&=4x+4\\ 4x-3x&=10 - 4\\ x&=6 \end{align*}$$

\]
Then \(AC=4x + 4=4\times6+4=28\).

Step4: Analyze the third - line segment problem (27)

Given \(MN+NP=MP\), where \(MN=x - 5\), \(NP = 3x+2\), and \(MP=6x - 23\). Set up the equation: \((x - 5)+(3x + 2)=6x - 23\).
\[

$$\begin{align*} x-5+3x + 2&=6x-23\\ 4x-3&=6x-23\\ 6x-4x&=23 - 3\\ 2x&=20\\ x&=10 \end{align*}$$

\]
Then \(NP=3x + 2=3\times10+2=32\).

Answer:

1. \(ST = 16\)
2. \(AC = 28\)
3. \(NP = 32\)