Question

1. compare corresponding sides that include the corresponding, congruent angles to show they are in proportion.

\\( \frac { a b } { a d } \\) is equal to
\\( \frac { a c } { a e } \\) is equal to

Explanation:

Step1: Calculate the length of \(AB\) and \(AD\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(AB\): \(A(3,1)\), \(B(1,4)\)
\(AB=\sqrt{(1 - 3)^2+(4 - 1)^2}=\sqrt{(- 2)^2+3^2}=\sqrt{4 + 9}=\sqrt{13}\)
For \(AD\): \(A(3,1)\), \(D(-1,7)\)
\(AD=\sqrt{(-1 - 3)^2+(7 - 1)^2}=\sqrt{(-4)^2+6^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}\)
Then \(\frac{AB}{AD}=\frac{\sqrt{13}}{2\sqrt{13}}=\frac{1}{2}\)

Step2: Calculate the length of \(AC\) and \(AE\)

For \(AC\): \(A(3,1)\), \(C(5,3)\)
\(AC=\sqrt{(5 - 3)^2+(3 - 1)^2}=\sqrt{2^2+2^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\)
For \(AE\): \(A(3,1)\), \(E(7,5)\)
\(AE=\sqrt{(7 - 3)^2+(5 - 1)^2}=\sqrt{4^2+4^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\)
Then \(\frac{AC}{AE}=\frac{2\sqrt{2}}{4\sqrt{2}}=\frac{1}{2}\)

Answer:

\(\frac{AB}{AD}\) is equal to \(\frac{1}{2}\).
\(\frac{AC}{AE}\) is equal to \(\frac{1}{2}\).