Can you find the missing side lengths of the Triangle? | (Olympiad Math) | #math #maths | #geometry

PreMath
10 Jun 202409:34

TLDRIn this video from Premath, a right triangle ABC with a perpendicular CD is presented, where side a is 169 units and the perpendicular CD is 60 units. The task is to find the missing side lengths b and c. By using the angle-angle similarity theorem and the Pythagorean triplets, the video demonstrates how to solve for the missing lengths, concluding with a and b being 25 and 144 units respectively, and the missing sides being 65 and 156 units.

Takeaways

  • 📐 The problem involves a right triangle ABC with a perpendicular CD, where side a is 169 units and the perpendicular CD is 60 units.
  • 🔍 The task is to find the missing side lengths b and c, as well as the length of the missing side a.
  • 📏 The video uses the Pythagorean theorem and angle-angle similarity to solve for the missing lengths.
  • 📐 Angles Alpha and Beta are complementary, summing up to 90°, which is key to establishing the similarity between triangles ACD and BCD.
  • ⚖️ By the angle-angle similarity theorem, the ratio of AD to CD is equal to the ratio of CD to BD, leading to the equation a * b = 60 * 60.
  • 🔢 The sum of segments a and b is equal to the total side length AB, which is 169 units.
  • 📉 The quadratic equation formed by the sum and product of roots is x² - 169x + 3600 = 0, which is derived from the given information.
  • 🧩 The constant 3600 is factored into 25 * 144, which helps in solving the quadratic equation by grouping.
  • 🔑 The roots of the quadratic equation are 25 and 144, representing the values of a and b respectively.
  • 📐 Using the Pythagorean triplets (5, 12, 13), the missing side lengths are found to be 65 and 156 units for triangles ACD and BCD respectively.
  • 🎉 The final answer is that segment AD is 25 units, BD is 144 units, the missing side c in triangle ACD is 65 units, and in triangle BCD is 156 units.

Q & A

  • What is the given side length 'a' of the right triangle ABC in the video?

    -The given side length 'a' of the right triangle ABC is 169 units.

  • What is the length of the perpendicular CD in the triangle?

    -The length of the perpendicular CD is 60 units.

  • What is the task presented in the video?

    -The task is to find the missing side lengths 'b' and 'c' of the triangle.

  • What is the relationship between angles Alpha and Beta in the triangle?

    -Angles Alpha and Beta are complementary, meaning their sum is equal to 90°.

  • Why are triangles ACD and BCD considered similar?

    -Triangles ACD and BCD are considered similar due to the Angle-Angle Similarity Theorem, as they both have a right angle and share angle Beta.

  • What is the proportion derived from the similarity of triangles ACD and BCD?

    -The proportion derived is that the ratio of AD to CD is equal to the ratio of CD to BD.

  • What is the equation formed by the cross-multiplication of the proportion from the similarity of triangles?

    -The equation formed is a * b = 60 * 60, which simplifies to 3600.

  • What is the sum of the segments a and b in the video?

    -The sum of the segments a and b is equal to 169 units.

  • How does the video solve for the missing side lengths using the quadratic equation?

    -The video uses the quadratic equation x² - (sum of roots) * x + (product of roots) = 0, with the sum of roots being 169 and the product of roots being 3600, to solve for the missing side lengths.

  • What are the final calculated values for the missing side lengths 'a' and 'b'?

    -The final calculated values are 'a' being 25 units and 'b' being 144 units.

  • How does the video use Pythagorean triplets to find the missing side lengths 'c'?

    -The video multiplies the Pythagorean triplets (5, 12, 13) by factors to match the calculated lengths of 'a' and 'b', resulting in new triplets that give the missing side lengths 'c' as 65 and 156 units.

Outlines

00:00

📚 Introduction to Right Triangle Problem

This paragraph introduces the problem of finding missing side lengths in a right triangle ABC with a perpendicular CD. The given side lengths are a=169 units and CD=60 units. The task is to find the lengths of sides b and c. The video script emphasizes the importance of subscribing and liking the video, and notes that the diagram may not be perfectly to scale. The first step involves labeling the segments and identifying complementary angles, Alpha and Beta, which sum up to 90°. The script then establishes the similarity between triangles ACD and BCD based on the angle-angle similarity theorem, leading to a proportion that will be used to solve for the missing lengths.

05:01

🔍 Solving the Triangle Using Proportions and Pythagorean Triplets

The second paragraph delves into solving the problem by setting up a proportion based on the sides of the triangles, leading to the equation a*b = 60*60 = 3600, which is labeled as equation number one. It also introduces equation number two, a + b = 169, which represents the total side length of AB. The script then transitions to using the quadratic equation formula, x² - (sum of roots)*x + (product of roots) = 0, with the sum of roots being 169 and the product being 3600. The constant 3600 is factored into 25*144, which corresponds to the sum of roots, 169. The quadratic is then solved by factoring, yielding two solutions for x, which are the roots a and b. The paragraph concludes by identifying the missing side lengths using Pythagorean triplets, multiplying the original triplet (5, 12, 13) by 5 and 12 to find the missing sides of 65 and 156 units, respectively. The video ends with a reminder to subscribe for more content.

Mindmap

Keywords

💡Right Triangle

A right triangle is a triangle that has one angle measuring exactly 90 degrees. In the context of the video, triangle ABC is a right triangle with a perpendicular line CD drawn from point C, creating two smaller right triangles, ACD and BCD. The video uses the properties of right triangles to solve for missing side lengths.

💡Perpendicular

Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. In the script, CD is described as a perpendicular line to AB, which is crucial for identifying the right angles in the triangles ACD and BCD and for applying the Pythagorean theorem.

💡Side Length

The term 'side length' refers to the measure of the distance between two endpoints of a line segment in a geometric figure. In the video, the side lengths of the triangle are essential for determining the missing lengths using the given values and geometric principles.

💡Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. In the script, angles Alpha and Beta are described as complementary, which is used to establish the similarity between triangles ACD and BCD.

💡Angle-Angle Similarity Theorem

The angle-angle similarity theorem states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. The video uses this theorem to establish the similarity between triangles ACD and BCD based on their corresponding angles.

💡Proportion

A proportion is an equation that states that two ratios are equal. In the video, the proportion is used to set up an equation relating the lengths of segments AD, CD, and BD, which is key to solving for the unknown side lengths.

💡Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically in the form of ax² + bx + c = 0. The video uses the quadratic equation to find the values of a and b, the unknown side lengths, by relating them to the sum and product of the roots.

💡Pythagorean Triplets

Pythagorean triplets are sets of three positive integers that satisfy the Pythagorean theorem (a² + b² = c²) for a right triangle. In the script, the presenter uses multiples of the basic Pythagorean triplet (5, 12, 13) to find the missing side lengths of the triangles ACD and BCD.

💡Sum of Roots

In the context of a quadratic equation, the sum of the roots is given by -b/a. The video uses this concept to relate the sum of the unknown side lengths (a and b) to the known total side length of the triangle AB (169 units).

💡Product of Roots

The product of the roots of a quadratic equation is given by c/a. In the video, the product of roots is used to establish the equation a * b = 3600, which helps in solving for the individual side lengths.

Highlights

Introduction to the problem of finding missing side lengths in a right triangle with given perpendicular.

Side length 'a' is given as 169 units, and the perpendicular 'CD' is 60 units.

The task is to find the missing side lengths 'b' and 'c'.

Assumption of complementary angles Alpha and Beta in the right triangle.

Identification of similar right triangles ACD and BCD using angle-angle similarity theorem.

Proportionality established between side lengths AD, CD, CD, and BD.

Equation 1: a * b = 3600 derived from the cross-multiplication.

Observation that side length AB equals 169, the sum of segments 'a' and 'b'.

Equation 2: a + b = 169 representing the sum of segments 'a' and 'b'.

Application of the quadratic equation formula to find the values of 'a' and 'b'.

Factorization of the constant 3600 into 25 * 144.

Transformation of the quadratic equation into x^2 - 169x + 3600 = 0.

Solving the quadratic equation by grouping and factoring to find the roots 25 and 144.

Conclusion that side length 'a' is 25 units and 'b' is 144 units.

Use of Pythagorean triplets to find the missing side lengths in triangles ACD and BCD.

Multiplication of Pythagorean triplets to match the given side lengths and find the missing ones.

Final determination of the missing side lengths as 65 units and 156 units.

Conclusion of the video with a summary of the found side lengths and a call to subscribe.

Create Stunning AI Art Instantly

No login. No cost. Just incredible AI images !

Create Stunning AI Art InstantlyGenerate Now