Question

7 calculate the altitude ah in the triangle below.

graph of triangle with points a(4,6), c(1,3), b(6,2) and a dashed line from a to h on the base

my calculations

grid for calculations

8 given the endpoints a(2,1), b(1,6) and c(6,2) are the vertices of triangle abc, calculate the length of altitude ah from vertex a.

my calculations

grid for calculations

9 calculate the height ah of the trapezoid below.

graph of trapezoid with points a(2,4), b(1,0), d(5,5), c(7,2) and h on the base

my calculations

grid for calculations

10 calculate the distance from the point a to the line shown on the graph below.

graph with point a(3,5) and a line from (0,3) to (4,-1)

my calculations

grid for calculations

Explanation:

Problem 7: Calculate the altitude \( AH \) in the triangle

Step 1: Identify coordinates of \( A, B, C, H \)

\( A(4,6) \), \( B(6,2) \), \( C(1,3) \). \( H \) lies on \( BC \); since \( AH \) is vertical? Wait, \( x \)-coordinate of \( A \) is 4, so \( H \) has \( x=4 \). Find \( y \)-coordinate of \( H \) (on \( BC \)).

Step 2: Equation of \( BC \)

Slope of \( BC \): \( m = \frac{2 - 3}{6 - 1} = \frac{-1}{5} \).
Equation: \( y - 3 = -\frac{1}{5}(x - 1) \).
Substitute \( x = 4 \): \( y - 3 = -\frac{1}{5}(3) \implies y = 3 - \frac{3}{5} = \frac{12}{5} = 2.4 \).

Step 3: Calculate \( AH \) (vertical distance)

\( AH = |6 - \frac{12}{5}| = |\frac{30 - 12}{5}| = \frac{18}{5} = 3.6 \).

Problem 8: Altitude \( AH \) in \( \triangle ABC \) with \( A(2,1) \), \( B(1,6) \), \( C(6,2) \)

Step 1: Equation of \( BC \)

Slope of \( BC \): \( m = \frac{2 - 6}{6 - 1} = \frac{-4}{5} \).
Equation: \( y - 6 = -\frac{4}{5}(x - 1) \implies 5y - 30 = -4x + 4 \implies 4x + 5y = 34 \).

Step 2: Distance from \( A(2,1) \) to \( BC \) (altitude formula)

Distance formula: \( d = \frac{|4(2) + 5(1) - 34|}{\sqrt{4^2 + 5^2}} = \frac{|8 + 5 - 34|}{\sqrt{41}} = \frac{|-21|}{\sqrt{41}} = \frac{21}{\sqrt{41}} \approx 3.28 \).

Problem 9: Height \( AH \) of the trapezoid

Step 1: Identify bases of trapezoid

Bases: \( AB \) (from \( A(2,4) \) to \( B(1,0) \))? No, trapezoid has \( B(1,0) \), \( H \), \( C(7,2) \), \( D(5,5) \). Wait, \( AH \) is vertical? \( A(2,4) \), \( H \) on \( BH \) (horizontal? \( B(1,0) \), \( H \) has \( y=0 \), \( x \) same as \( A \)? \( A(2,4) \), so \( H(2,0) \).

Step 2: Calculate \( AH \)

Vertical distance: \( |4 - 0| = 4 \).

Problem 10: Distance from \( A(3,5) \) to the line

Answer:

s:

1. \( \boldsymbol{\frac{18}{5} = 3.6} \)
2. \( \boldsymbol{\frac{21}{\sqrt{41}} \approx 3.28} \) (or rationalized \( \frac{21\sqrt{41}}{41} \))
3. \( \boldsymbol{4} \)
4. \( \boldsymbol{\frac{5\sqrt{2}}{2} \approx 3.54} \)