Calculate AE in Triangle: Parallel Lines with Segments 6 and 10

Q: How do I know when triangles are similar?

When a line is parallel to one side of a triangle, it creates a smaller similar triangle. The angles are equal due to parallel line properties, making the triangles similar.

Q: What does it mean for ratios to be consistent?

In similar triangles, all corresponding sides must have the same ratio. If $ \frac{AD}{AB} \neq \frac{DE}{BC} $, the measurements contradict each other.

Q: Why can't I just solve for AE anyway?

Because the given measurements are mathematically impossible! If you solve anyway, you'll get a number that doesn't represent reality. Always check consistency first.

Q: How do I check if measurements are consistent?

Calculate the ratio of corresponding sidesSimplify each fractionCompare: if ratios are equal, solve; if not, the figure is impossible

Q: What should I do when I find inconsistent data?

State that "the figure cannot exist" or is impossible. This shows you understand the mathematical relationship and can identify contradictory information.

Q: Is this type of problem common in real tests?

Yes! Many geometry problems test whether you can identify impossible figures rather than just calculate. It's an important critical thinking skill in mathematics.

Similar Triangles with Inconsistent Data

BC is parallel to DE.

Calculate AE.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find the length of AE.

00:08 We have parallel lines as shown in the given data.

00:12 Remember, corresponding angles between parallel lines are equal.

00:17 The triangles share the same vertex angle as given.

00:21 This means the triangles are similar by the angle-angle rule.

00:29 Next, we'll find the similarity ratio.

00:37 Now, substitute the appropriate values and solve for AE.

00:45 Remember, the whole side equals the sum of its parts.

00:53 Here, the similarity ratio does not match, making the drawing incorrect.

00:58 And that's how we solve the problem. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

BC is parallel to DE.

Calculate AE.

Step-by-step solution

Let's prove that triangles ADE and ABC are similar using:

Since DE is parallel to BC, angles ADE and ABC are equal (according to the law - between parallel lines, corresponding angles are equal)

Angle DAE and angle BAC are equal since it's the same angle

After we proved that the triangles are similar, let's write the given data from the drawing according to the following similarity ratio:

$\frac{AD}{AB}=\frac{DE}{BC}=\frac{AE}{AC}$

We know that - $AB=AD+DB=6+4=10$

$\frac{6}{10}=\frac{10}{154}=\frac{AE}{AC}$

Let's reduce the fractions:

$\frac{3}{5}=\frac{2}{3}$

This statement is incorrect, meaning the data in the drawing contradicts the fact that the triangles are similar. Therefore, the drawing is impossible.

Final Answer

Impossible as the shape in the figure cannot exist.

Key Points to Remember

Essential concepts to master this topic

Similarity Rule: When lines are parallel, corresponding triangles must be similar
Ratio Check: Calculate $\frac{AD}{AB} = \frac{6}{10} = \frac{3}{5}$ and $\frac{DE}{BC} = \frac{10}{15} = \frac{2}{3}$
Verification: Compare ratios: if $\frac{3}{5} \neq \frac{2}{3}$ , the figure is impossible ✓

Common Mistakes

Avoid these frequent errors

Calculating AE without checking if the figure is possible
Don't just set up proportion equations and solve for AE = impossible answer! When parallel lines create similar triangles, ALL corresponding ratios must be equal. Always verify that the given measurements create consistent ratios before solving.

Practice Quiz

Test your knowledge with interactive questions

If it is known that both triangles are equilateral, are they therefore similar?

FAQ

Everything you need to know about this question

How do I know when triangles are similar?

When a line is parallel to one side of a triangle, it creates a smaller similar triangle. The angles are equal due to parallel line properties, making the triangles similar.

What does it mean for ratios to be consistent?

In similar triangles, all corresponding sides must have the same ratio. If $\frac{AD}{AB} \neq \frac{DE}{BC}$ , the measurements contradict each other.

Why can't I just solve for AE anyway?

Because the given measurements are mathematically impossible! If you solve anyway, you'll get a number that doesn't represent reality. Always check consistency first.

How do I check if measurements are consistent?

Calculate the ratio of corresponding sides
Simplify each fraction
Compare: if ratios are equal, solve; if not, the figure is impossible

What should I do when I find inconsistent data?

State that "the figure cannot exist" or is impossible. This shows you understand the mathematical relationship and can identify contradictory information.

Is this type of problem common in real tests?

Yes! Many geometry problems test whether you can identify impossible figures rather than just calculate. It's an important critical thinking skill in mathematics.

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