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.