Triangle-Angle-Bisector Theorem

Summary

This video script explores the Triangle Angle Bisector Theorem, which states that a ray bisecting an angle of a triangle divides the opposite side into segments proportional to the other two sides. The example demonstrates this by using a diagram where ray GK bisects angle IGH. By setting up the proportion as IK/KH = IG/HG and substituting known values, the script guides viewers through solving for the unknown segment length, finding x to be 18. The explanation is clear and engages viewers in understanding the theorem's practical application.

Takeaways

• 📐 The Triangle Angle Bisector Theorem states that if a ray bisects an angle of a triangle, it divides the opposite side into two segments that are proportional to the other two sides.
• 📏 In the given diagram, angle 'i gh' is bisected by line 'g k', which means 'ih' is proportional to the other sides of the triangle.
• ✂️ The proportion is expressed as 'ik/kh' being equal to 'ig/hg', which is a direct application of the Triangle Angle Bisector Theorem.
• 🔍 The values for 'ik' and 'kh' are substituted with variables and known lengths, respectively, to set up a proportion equation.
• 📝 The substitution results in the equation 'ik/30 = 24/40', which is used to solve for the unknown segment 'ik'.
• 🧮 Cross-multiplication is used to solve the proportion, resulting in the equation '40 * x = 30 * 24'.
• 🔢 Simplifying the equation by dividing both sides by 40 gives 'x = 720 / 40', which simplifies to 'x = 18'.
• 📈 The solution 'x = 18' represents the length of segment 'ik', demonstrating the practical use of the Triangle Angle Bisector Theorem.
• 📚 The script provides a clear example of how to apply the Triangle Angle Bisector Theorem to find the length of a segment in a triangle.
• 📘 The process involves setting up a proportion, substituting known values, using cross-multiplication, and simplifying to find the unknown segment length.
• 📙 The example concludes with the final answer, reinforcing the understanding of the theorem and its application in geometry.

Q & A

What is the Triangle Angle Bisector Theorem?

The Triangle Angle Bisector Theorem states that if a ray bisects an angle of a triangle, it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

What does it mean for a ray to bisect an angle in a triangle?

When a ray bisects an angle in a triangle, it divides the angle into two equal angles.

In the given diagram, which angle is bisected by ray GK?

In the given diagram, the angle IGH is bisected by ray GK.

What is the relationship between the segments IH and KH when a ray bisects an angle in a triangle?

When a ray bisects an angle in a triangle, the segment opposite to the bisected angle (IH) is proportional to the other two sides of the triangle (KH and IG).

How is the proportion between the segments IH and KH expressed mathematically?

The proportion between the segments IH and KH is expressed as IK/KH = IG/HG.

What values are substituted for the segments in the proportion to solve for x?

In the proportion, x is substituted for IK, 30 for KH, 24 for IG, and 40 for HG.

What is the cross product of the segments in the proportion?

The cross product in the proportion is 40 times x (40x) and 30 times 24 (720).

How is the value of x determined from the cross product?

The value of x is determined by dividing the cross product (720) by the known side (40), resulting in x = 18.

What is the final value of x in the proportion?

The final value of x, which represents the length of segment IK, is 18.

How does the Triangle Angle Bisector Theorem help in solving the given problem?

The Triangle Angle Bisector Theorem helps in solving the problem by establishing a proportion between the segments of the triangle, allowing us to find the unknown segment length using the known lengths.

Outlines

00:00

📐 Triangle Angle Bisector Theorem Application

This paragraph explains the application of the Triangle Angle Bisector Theorem, which states that a ray bisecting an angle of a triangle divides the opposite side into two segments that are proportional to the other two sides. The example provided uses a diagram with a bisected angle and congruent sides marked. The proportion is set up with the formula 'ik/kh = ig/hg', where 'ik' and 'kh' are the segments of the divided side, and 'ig' and 'hg' are the lengths of the other two sides. The values are substituted into the proportion, resulting in a cross-multiplication that leads to the calculation of 'x' as 18, which represents the length of 'ik'.

Keywords

💡Triangle Angle Bisector Theorem

The Triangle Angle Bisector Theorem is a fundamental principle in geometry that states if a ray bisects an angle of a triangle, it divides the opposite side into two segments that are proportional to the lengths of the other two sides of the triangle. In the video, this theorem is the central concept used to demonstrate how to find the length of a segment in a triangle when the bisector of an angle is known. The example given in the script illustrates the theorem by showing that the ratio of the lengths of the segments created by the bisector is equal to the ratio of the lengths of the other two sides of the triangle.

💡Ray

A ray in geometry is a line that starts at a certain point and extends infinitely in one direction. In the context of the video, the ray 'gk' bisects the angle 'i gh', which means it divides the angle into two equal parts. This is important for applying the Triangle Angle Bisector Theorem because the ray's action of bisecting the angle is what allows for the proportional division of the opposite side.

💡Congruence

Congruence in geometry refers to the property of two figures being identical in shape and size. In the script, the congruence of the sides 'ig' and 'hg' is marked, indicating that they are of equal length. This is a key piece of information because it allows the use of the Triangle Angle Bisector Theorem to establish a proportion involving the segments created by the angle bisector.

💡Proportion

Proportion in mathematics is a relationship between two ratios that are equal. In the video, the proportion is established by the Triangle Angle Bisector Theorem, where the ratio of the lengths of the segments 'ik' to 'kh' is equal to the ratio of 'ig' to 'hg'. This proportion is used to solve for the unknown length 'x' in the example provided.

💡Segments

In geometry, a segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. In the script, the term 'segments' refers to the parts of the side 'ih' that are created by the angle bisector 'gk'. The lengths of these segments are related through the proportion established by the Triangle Angle Bisector Theorem.

💡Substitute

To substitute in mathematics means to replace a variable or an unknown with a known value or expression. In the video, the values for 'ik', 'kh', 'ig', and 'hg' are substituted into the proportion to solve for 'x', which represents the length of the segment 'ik'. This substitution is a critical step in applying the theorem to find the unknown segment length.

💡Cross Products

Cross products in the context of this video refer to the multiplication of the lengths of the segments to create expressions that can be equated in the proportion. For example, '40 times x' and '30 times 24' are the cross products used to set up the equation that will be solved for 'x'. This step is part of the algebraic process of applying the proportion to find the unknown length.

💡Divide

In mathematics, to divide means to partition a quantity into a number of equal parts. In the script, dividing each side of the equation by 40 is the method used to isolate 'x' and find its value. This is a standard algebraic technique for solving equations.

💡Example

An example in an educational context is a specific case or scenario used to illustrate a concept or principle. In the video, 'example three' is used to demonstrate the application of the Triangle Angle Bisector Theorem with specific values for the sides of the triangle. This example helps to clarify how the theorem is used in practice.

💡Lengths

Lengths in geometry refer to the measure of the distance between two points along a line. The script discusses the lengths of various segments within a triangle, specifically focusing on how the Triangle Angle Bisector Theorem relates the lengths of these segments. The lengths of 'ig', 'hg', 'ik', and 'kh' are central to the example problem presented in the video.

💡Algebraic Process

The algebraic process involves using algebraic techniques to solve for unknowns in an equation or set of equations. In the video, the algebraic process includes setting up a proportion based on the Triangle Angle Bisector Theorem, substituting known values, and then solving for the unknown length 'x' by using cross products and division. This process is key to finding the solution to the example problem.

Highlights

Introduction to the Triangle Angle Bisector Theorem

Explanation of the theorem's principle: a ray bisecting an angle divides the opposite side into proportional segments

Diagram analysis: angle igh is bisected by ray gk

Congruence of sides marked, indicating proportionality of side ih to the other two sides

Setting up the proportion: ik/kh = ig/hg

Substituting known values into the proportion equation

Assigning x to side ik, 30 to kh, 24 to ig, and 40 to hg

Calculating cross products: 40x and 30 * 24

Simplifying the equation by dividing both sides by 40

Solving for x, finding x equals 18

Demonstration of the Triangle Angle Bisector Theorem in practice

Application of the theorem to a specific triangle diagram

Step-by-step guide to solving a problem using the theorem

Importance of understanding the theorem for geometric problem-solving

Visual representation aiding in the comprehension of the theorem

Use of algebraic methods to solve geometric problems

Conclusion of the example problem using the Triangle Angle Bisector Theorem

Transcripts

using the triangle angle bisector

theorem

remember

the triangle angle bisector theorem says

if a ray bisects an angle of a triangle

then it divides the opposite side into

two segments that are proportional to

the other two sides of the triangle

for this diagram this big angle i gh is

bisected by

g k

we know that because they are marked

congruent that means

this side

ih

is going to be proportional

to the other two sides of the triangle

so ik

over kh

will be equal to ig

over hg

so let's write that

out

ik

over kh

is going to be equal to

ig

over hg

now that we have

our proportion

with the measures of the signs or the

ends of points of the sides we can

substitute in the values and for ik

we'll substitute x

and for kh we'll substitute 30.

and for ig we'll substitute 24

and for hg we'll substitute 40.

use our cross products 40 times x

will give 40x

and 30 times 24 will give 720.

divide each side of that by 40

x is equal to 18.

so there's example three using the

triangle angle bisector theorem