of the orthocenter are (1, 3).
triangle fgh has coordinates f(-3, 1), g(2, 6), and h(4, 1).
9. find an equation of the line containing the
altitude from g to fh.

10. find an equation of the line containing the
altitude from h to fg.

11. solve the system of equations from exercises 9
and 10 to find the coordinates of the orthocenter.

find the orthocenter of the triangle with the given vertices.

Question

of the orthocenter are (1, 3).
triangle fgh has coordinates f(-3, 1), g(2, 6), and h(4, 1).
9. find an equation of the line containing the
   altitude from g to fh.

10. find an equation of the line containing the
    altitude from h to fg.

11. solve the system of equations from exercises 9
    and 10 to find the coordinates of the orthocenter.

find the orthocenter of the triangle with the given vertices.

Accepted Answer

### Problem 9:
# Explanation:
## Step1: Find slope of FH
Points $ F(-3,1) $ and $ H(4,1) $. Slope $ m_{FH} = \frac{1 - 1}{4 - (-3)} = 0 $. So $ FH $ is horizontal.
## Step2: Determine altitude slope
Altitude from $ G $ to $ FH $ is vertical (perpendicular to horizontal). So slope is undefined.
## Step3: Equation of altitude
Passes through $ G(2,6) $, vertical line: $ x = 2 $.
# Answer: $ x = 2 $

### Problem 10:
# Explanation:
## Step1: Find slope of FG
Points $ F(-3,1) $ and $ G(2,6) $. Slope $ m_{FG} = \frac{6 - 1}{2 - (-3)} = \frac{5}{5} = 1 $.
## Step2: Determine altitude slope
Altitude from $ H $ to $ FG $ is perpendicular, so slope $ m = -1 $ (negative reciprocal).
## Step3: Equation using point-slope
Using $ H(4,1) $: $ y - 1 = -1(x - 4) $ → $ y = -x + 5 $.
# Answer: $ y = -x + 5 $

### Problem 11:
# Explanation:
## Step1: System of equations
From 9: $ x = 2 $; From 10: $ y = -x + 5 $.
## Step2: Substitute $ x = 2 $
Into $ y = -2 + 5 = 3 $.
# Answer: $ (2, 3) $ (Wait, correction: Wait, in problem 9, the altitude from G to FH: FH is horizontal (y=1), so altitude is vertical through G(2,6), so x=2. Then problem 10: altitude from H to FG: slope of FG is 1, so altitude slope -1, equation $ y - 1 = -1(x - 4) $ → $ y = -x + 5 $. Then solve $ x = 2 $ and $ y = -2 + 5 = 3 $. Wait, but the initial note said orthocenter (1,3)? Wait, maybe miscalculation. Wait, let's recheck problem 9: F(-3,1), H(4,1): FH is horizontal (y=1), so altitude from G(2,6) to FH is vertical line x=2? Wait, no: altitude is perpendicular to FH and passes through G. Since FH is horizontal (slope 0), altitude is vertical (undefined slope), so yes, x=2. Then problem 10: slope of FG: F(-3,1), G(2,6): (6-1)/(2+3)=1, so altitude from H(4,1) to FG: slope -1, equation y -1 = -1(x -4) → y = -x +5. Then intersection of x=2 and y=-x+5: x=2, y= -2 +5=3. Wait, but the initial text said "of the orthocenter are (1, 3)" – maybe a typo. Wait, maybe my problem 9 is wrong? Wait, no: FH is from (-3,1) to (4,1), so it's horizontal line y=1. The altitude from G to FH is a vertical line? No, wait: altitude is a segment from G to FH, perpendicular to FH. Since FH is horizontal, the altitude is vertical? Wait, no: horizontal line has slope 0, so perpendicular is vertical (undefined slope), so yes, the altitude is vertical line through G(2,6), so x=2. Then problem 10: slope of FG is 1, so altitude from H is slope -1, equation y= -x +5. Then solving x=2 and y= -x +5: x=2, y=3. So orthocenter (2,3). But the initial note said (1,3) – maybe a mistake in the note. So the solution here is (2,3).)

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QUESTION IMAGE

Question

Explanation:

Problem 9:

Step1: Find slope of FH

Step2: Determine altitude slope

Step3: Equation of altitude

Step1: Find slope of FG

Step2: Determine altitude slope

Step3: Equation using point-slope

Step1: System of equations

Step2: Substitute \( x = 2 \)

Answer:

Problem 10: