Question

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use graph paper to plot the triangle, then answer the questions.
8 the vertices of triangle abc are: a(-4,4) b(-7,9) c(1,7)
(a)
complete the steps below to prove whether triangle abc is an isosceles right triangle.
find the slopes of the following:
\\( \overline { a b } = \\)
\\( \overline { b c } = \\)
\\( \overline { a c } = \\)
find the lengths of the following:
\\( \overline { a b } = \\)
\\( \overline { b c } = \\)
\\( \overline { a c } = \\)

Explanation:

Step1: Calculate slope of \( \overline{AB} \)

Slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For \( A(-4,4) \) and \( B(-7,9) \), \( m_{AB}=\frac{9 - 4}{-7+4}=\frac{5}{-3}=-\frac{5}{3} \)

Step2: Calculate slope of \( \overline{BC} \)

For \( B(-7,9) \) and \( C(1,7) \), \( m_{BC}=\frac{7 - 9}{1 + 7}=\frac{-2}{8}=-\frac{1}{4} \)

Step3: Calculate slope of \( \overline{AC} \)

For \( A(-4,4) \) and \( C(1,7) \), \( m_{AC}=\frac{7 - 4}{1+4}=\frac{3}{5} \)

Step4: Calculate length of \( \overline{AB} \)

Distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). For \( A(-4,4) \) and \( B(-7,9) \), \( d_{AB}=\sqrt{(-7 + 4)^2+(9 - 4)^2}=\sqrt{9 + 25}=\sqrt{34} \)

Step5: Calculate length of \( \overline{BC} \)

For \( B(-7,9) \) and \( C(1,7) \), \( d_{BC}=\sqrt{(1 + 7)^2+(7 - 9)^2}=\sqrt{64+4}=\sqrt{68}=2\sqrt{17} \)

Step6: Calculate length of \( \overline{AC} \)

For \( A(-4,4) \) and \( C(1,7) \), \( d_{AC}=\sqrt{(1 + 4)^2+(7 - 4)^2}=\sqrt{25 + 9}=\sqrt{34} \)

Answer:

Slopes: \( \overline{AB}=-\frac{5}{3} \), \( \overline{BC}=-\frac{1}{4} \), \( \overline{AC}=\frac{3}{5} \)
Lengths: \( \overline{AB}=\sqrt{34} \), \( \overline{BC}=2\sqrt{17} \), \( \overline{AC}=\sqrt{34} \)