Question

triangles abc and ade are similar, as shown. which must be true? check all that apply.
ac = ab
\\( \frac { a c } { a e } = \frac { b c } { d e } \\)
\\( \frac { a c } { a d } = \frac { a b } { a e } \\)
de = 2bc
bc = 2de
ac = 2ae
ae = 2ac

Explanation:

Step1: Recall the property of similar triangles

For similar triangles \( \triangle ABC\) and \( \triangle ADE\), the ratios of corresponding sides are equal. That is \(\frac{AB}{AD}=\frac{AC}{AE}=\frac{BC}{DE}\).

Step2: Analyze each option

• Option \(AC = AB\):

From the graph, \(AB\) is the length from \(A(0,0)\) to \(B(- 1,2)\), using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(AB=\sqrt{(-1 - 0)^2+(2 - 0)^2}=\sqrt{1 + 4}=\sqrt{5}\). \(AC\) is the length from \(A(0,0)\) to \(C(2,2)\), \(AC=\sqrt{(2 - 0)^2+(2 - 0)^2}=\sqrt{4 + 4}=2\sqrt{2}\). So \(AC
eq AB\).

• Option \(\frac{AC}{AE}=\frac{BC}{DE}\):

Since \(\triangle ABC\sim\triangle ADE\), by the property of similar - triangles (corresponding sides are proportional), this is True.

• Option \(\frac{AC}{AD}=\frac{AB}{AE}\):

From \(\frac{AB}{AD}=\frac{AC}{AE}\), cross - multiplying gives \(AB\times AE=AC\times AD\), or \(\frac{AC}{AD}=\frac{AB}{AE}\). This is True.

• Option \(DE = 2BC\):

\(BC\) is the length from \(B(-1,2)\) to \(C(2,2)\), \(BC=\vert2-(-1)\vert = 3\). \(DE\) is the length from \(D(-2,-4)\) to \(E(4,-4)\), \(DE=\vert4-(-2)\vert=6\). So \(DE = 2BC\). This is True.

• Option \(BC = 2DE\):

Since \(BC = 3\) and \(DE = 6\), \(BC=\frac{1}{2}DE\), so \(BC
eq2DE\).

• Option \(AC = 2AE\):

\(AC = 2\sqrt{2}\), \(AE\) is the length from \(A(0,0)\) to \(E(4,-4)\), \(AE=\sqrt{(4 - 0)^2+(-4 - 0)^2}=\sqrt{16 + 16}=4\sqrt{2}\). So \(AE = 2AC\), not \(AC = 2AE\).

• Option \(AE = 2AC\):

Since \(AC = 2\sqrt{2}\) and \(AE=4\sqrt{2}\), \(AE = 2AC\). This is True.

Answer:

\(\frac{AC}{AE}=\frac{BC}{DE}\), \(\frac{AC}{AD}=\frac{AB}{AE}\), \(DE = 2BC\), \(AE = 2AC\)