In the figure, $A C$ and $D E$ are the angle bisectors of $\angle B A D$ and $\angle A D B$ respectively. If $A B=2 B C$ and $A D=B D$ , find $A E: C D$ . $\newline$ (A) $1: 1$ $\newline$ (B) $1: 2$ $\newline$ (C) $2: 1$ $\newline$ (D) $D E$ $0$

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Algebra 1

Compare linear and exponential growth

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Q. In the figure,

A C

and

D E

are the angle bisectors of

\angle B A D

and

\angle A D B

respectively. If

A B=2 B C

and

A D=B D

, find

A E: C D

\newline

(A)

1: 1

\newline

(B)

1: 2

\newline

(C)

2: 1

\newline

(D)

D E

0

Identify Given Information: Identify the given information and the properties of angle bisectors in a triangle. $\newline$ Given: $AB = 2BC$ and $AD = BD$ (which means triangle $ABD$ is isosceles). $\newline$ The angle bisector theorem states that the ratio of the lengths of the two segments created on the base by an angle bisector is proportional to the ratio of the lengths of the other two sides of the triangle.
Apply Angle Bisector Theorem (ABD): Apply the angle bisector theorem to triangle $ABD$ , where $DE$ is the angle bisector. $\newline$ According to the angle bisector theorem, $\frac{AE}{ED} = \frac{AB}{BD}$ . $\newline$ Since $AD = BD$ , $AB = 2BD$ , so $\frac{AE}{ED} = \frac{2}{1}$ .
Apply Angle Bisector Theorem (ABC): Apply the angle bisector theorem to triangle ABC, where $AC$ is the angle bisector. $\newline$ According to the angle bisector theorem, $\frac{CD}{DA} = \frac{CB}{BA}$ . $\newline$ Since $AB = 2BC$ , $\frac{CD}{DA} = \frac{BC}{2BC} = \frac{1}{2}$ .
Combine Ratios: Combine the ratios obtained from the angle bisector theorem to find $AE:CD$ . From step $2$ , we have $AE = 2ED$ . From step $3$ , we have $CD = \frac{1}{2} DA$ . Since $AD = BD$ , we can substitute $DA$ with $BD$ in the ratio for $CD$ . So, $CD = \frac{1}{2} BD$ .
Find $AE:CD$ : Since $AE = 2ED$ and $CD = \frac{1}{2} BD$ , and knowing that $AD = BD$ , we can find the ratio $AE:CD$ by comparing $AE$ to $BD$ and $CD$ to $BD$ . $AE:BD = 2ED:BD = 2:1$ (since $AE = 2ED$ and $AE = 2ED$ $1$ is part of $BD$ ). $CD:BD = \frac{$1$}{$2$} BD:BD = \frac{$1$}{$2$}:$1$.
Compare Ratios: To find $AE:CD$, we need to compare $AE$ to $CD$ directly.$\newline$We have $AE:BD = 2:1$ and $CD:BD = \frac{1}{2}:1$.$\newline$To compare $AE$ to $CD$, we need to have the same reference for $BD$ in both ratios.$\newline$We can multiply the second ratio by $2$ to have the same reference for $BD$.$\newline$So, $AE:BD = 2:1$ and $AE$$1$.
Find $AE:CD$: Now that we have $AE:BD$ and $CD:BD$ with the same reference, we can find $AE:CD$. $\newline$$AE:CD = (AE:BD) / (CD:BD) = (2:1) / (1:2) = \frac{2}{1} \times \frac{2}{1} = 4:1$.