Line segments AC and BD intersect at point E such that segment AB is parallel to segment C… — SAT

Line segments $AC$ and $BD$ intersect at point $E$ such that segment $AB$ is parallel to segment $CD$ . If the length of $AE$ is $5$ , the length of $CE$ is $10$ , the length of $BE$ is $x - 2$ , and the length of $DE$ is $x + 4$ , what is the value of $x$ ?

Cevap: 8

Cevap

Since segment

AB

is parallel to segment

CD

, the alternate interior angles

\angle EAB

and

\angle ECD

are congruent, and vertical angles

\angle AEB

and

\angle CED

are congruent. By the Angle-Angle (AA) similarity theorem, triangle

ABE

is similar to triangle

CDE

. The ratio of the lengths of corresponding sides is equal, so

\frac{AE}{CE} = \frac{BE}{DE}

. Substituting the given lengths gives

\frac{5}{10} = \frac{x - 2}{x + 4}

. Simplifying the left side to

\frac{1}{2}

and cross-multiplying gives

x + 4 = 2(x - 2)

, which expands to

x + 4 = 2x - 4

. Solving for

x

yields

x = 8

Adım Adım Çözüm

Establish the similarity of triangles

ABE

and

CDE

\triangle ABE \sim \triangle CDE

Since segment

AB

is parallel to segment

CD

, alternate interior angles

\angle EAB

and

\angle ECD

are congruent, and vertical angles

\angle AEB

and

\angle CED

are congruent. Thus, the triangles are similar by AA similarity.

Set up a proportion using the ratio of corresponding sides.

\frac{AE}{CE} = \frac{BE}{DE}

In similar triangles, the ratio of corresponding side lengths is constant.

Substitute the given algebraic expressions and segment lengths into the proportion.

\frac{5}{10} = \frac{x - 2}{x + 4}

The given values are

AE = 5

CE = 10

BE = x - 2

, and

DE = x + 4

Simplify the fraction and solve the linear equation for

x

x = 8

Simplifying

\frac{5}{10}

yields

\frac{1}{2}

. Cross-multiplying gives

1 \cdot (x + 4) = 2 \cdot (x - 2)

, which simplifies to

x + 4 = 2x - 4

. Subtracting

x

from both sides and adding

4

to both sides gives

x = 8

Anahtar Kavram

Triangle similarity criteria (specifically AA similarity) and using proportions of corresponding sides in similar triangles to solve for unknown variables.

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Adım Adım Çözüm

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