Trisection and Congruent Triangles

Oct 13, 2025 by wbrum.org 137 views

Introduction

In the realm of geometry, Olympiad problems often present a unique blend of mathematical concepts and logical reasoning. The problem at hand, involving the trisection of a triangle, is a classic example of such a conundrum. By carefully analyzing the given conditions and applying geometric principles, we can unravel the mystery behind this seemingly complex problem.

Problem Statement

Let $D$ and $E$ be two points on the side $AC$ of $\triangle ABC$ such that $A, D, E, C$ are in that order and $\angle ABD = \angle DBE = \angle EBC = 45^\circ$ . If $AB + BE = 10$ and $DB + BC = 15$ , we are tasked with finding the value of $AC$ .

Geometric Analysis

To approach this problem, we must first understand the geometric implications of the given conditions. The fact that $\angle ABD = \angle DBE = \angle EBC = 45^\circ$ suggests that the triangle is divided into three congruent segments. This is a classic example of a trisection, where a line segment is divided into three equal parts.

Trisection and Congruent Triangles

By drawing the altitude from point $B$ to side $AC$ , we create two right-angled triangles: $\triangle ABD$ and $\triangle BDC$ . Since $\angle ABD = \angle DBE = \angle EBC = 45^\circ$ , we can conclude that these triangles are congruent.

Congruent Triangles and Side Lengths

Using the properties of congruent triangles, we can establish relationships between the side lengths of $\triangle ABD$ and $\triangle BDC$ . Specifically, we know that the corresponding sides of congruent triangles are equal in length.

Applying the Pythagorean Theorem

To further analyze the problem, we can apply the Pythagorean Theorem to the right-angled triangles $\triangle ABD$ and $\triangle BDC$ . This will allow us to establish relationships between the side lengths and ultimately find the value of $AC$ .

Pythagorean Theorem and Side Lengths

Applying the Pythagorean Theorem to $\triangle ABD$ , we get:

AB^2 + BD^2 = AD^2

Similarly, applying the Pythagorean Theorem to $\triangle BDC$ , we get:

BC^2 + BD^2 = DC^2

Substituting and Simplifying

We can substitute the given values of $AB + BE = 10$ and $DB + BC = 15$ into the equations above. This will allow us to simplify the expressions and ultimately find the value of $AC$ .

Solving for AC

After simplifying the expressions, we can solve for the value of $AC$ . This will involve combining the equations and using algebraic manipulations to isolate the variable $AC$ .

Conclusion

In conclusion, the triangle's trisection Olympiad problem is a classic example of a geometric conundrum that requires careful analysis and application of mathematical principles. By understanding the geometric implications of the given conditions and applying the Pythagorean Theorem, we can unravel the mystery behind this seemingly complex problem and find the value of $AC$ .

Solution

Let's denote $AB = x$ and $BC = y$ . Then, we have:

x + BE = 10

DB + y = 15

Since $\triangle ABD$ and $\triangle BDC$ are congruent, we have:

AB = BD = x

BC = DC = y

Applying the Pythagorean Theorem to $\triangle ABD$ , we get:

x^2 + x^2 = AD^2

Similarly, applying the Pythagorean Theorem to $\triangle BDC$ , we get:

y^2 + y^2 = DC^2

Substituting the given values, we get:

x + BE = 10

x + y = 15

Solving for $x$ and $y$ , we get:

x = 5

y = 10

Now, we can find the value of $AC$ :

AC = AD + DC = 2x + 2y = 2(5) + 2(10) = 30

Therefore, the value of $AC$ is $\boxed{30}$ .

Final Thoughts

Introduction

In our previous article, we explored the triangle's trisection Olympiad problem, a classic example of a geometric conundrum that requires careful analysis and application of mathematical principles. In this article, we will delve deeper into the problem and provide a Q&A section to help clarify any doubts or questions you may have.

Q&A

Q: What is the significance of the 45-degree angles in the problem?

A: The 45-degree angles are crucial in establishing the congruence of the triangles. By having three congruent angles, we can conclude that the triangles are similar and have proportional side lengths.

Q: Why is the Pythagorean Theorem used in the solution?

A: The Pythagorean Theorem is used to establish relationships between the side lengths of the triangles. By applying the theorem to the right-angled triangles, we can find the values of the side lengths and ultimately determine the value of AC.

Q: Can you explain the concept of congruent triangles in more detail?

A: Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides of congruent triangles are equal in length, and the corresponding angles are equal in measure.

Q: How do we know that the triangles are congruent?

A: We know that the triangles are congruent because they have three congruent angles. This is a sufficient condition for two triangles to be congruent.

Q: What is the relationship between the side lengths of the triangles?

A: The side lengths of the triangles are related by the Pythagorean Theorem. By applying the theorem to the right-angled triangles, we can find the values of the side lengths and ultimately determine the value of AC.

Q: Can you provide a step-by-step solution to the problem?

A: Here is a step-by-step solution to the problem:

Denote AB = x and BC = y.
We have x + BE = 10 and DB + y = 15.
Since ∆ABD and ∆BDC are congruent, we have AB = BD = x and BC = DC = y.
Applying the Pythagorean Theorem to ∆ABD, we get x^2 + x^2 = AD^2.
Similarly, applying the Pythagorean Theorem to ∆BDC, we get y^2 + y^2 = DC^2.
Substituting the given values, we get x + BE = 10 and x + y = 15.
Solving for x and y, we get x = 5 and y = 10.
Now, we can find the value of AC: AC = AD + DC = 2x + 2y = 2(5) + 2(10) = 30.

Q: What is the significance of the trisection in the problem?

A: The trisection is significant because it allows us to divide the triangle into three congruent segments. This is a key concept in the solution to the problem.

Q: Can you provide any additional tips or insights for solving the problem?

A: Here are some additional tips and insights for solving the problem:

Make sure to carefully read and understand the problem statement.
Use geometric reasoning and visualization to help solve the problem.
Apply the Pythagorean Theorem and other mathematical principles as needed.
Check your work and make sure that your solution is correct.