MHB Find Length of Side in Right Triangle w/ 45°, 90°, and 45° Angle

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In a right triangle ABC with a right angle at A and angle bisector BD, the lengths AB and BC are given as 12 cm and 15 cm, respectively. To find the length of segment AD, the Angle Bisector Theorem is applied, leading to the equation AD/12 = (9 - AD)/15. Solving this equation reveals that AD equals 6 cm. The discussion highlights the importance of understanding the Angle Bisector Theorem, especially for students who have not yet learned trigonometry. This approach provides a foundational method for solving problems involving right triangles and angle bisectors.
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The triangle ABC is a right triangle with A as the right angle and BD is the bisector of angle B. If AB = 12 cm and BC = 15 cm, the length of AD is ...
A. 3 cm
B. 4 cm
C. 5 cm
D. 6 cm
It was a question for a 9th grader and the book hasn't covered trigonometry by name yet (As in, they don't know about the term sine, cosine, and tangent, but the books do explain the length ratio of triangle which has 45°-90°-45° angle or 30°-60°-90° angle. How to do it and explain it to them?
 

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Pythagoras $\implies AC = 9$

using the ...

Angle Bisector Theorem

$\dfrac{AD}{AB} = \dfrac{CD}{BC}$

$\dfrac{AD}{12} = \dfrac{9-AD}{15}$

solve for $AD$
 
skeeter said:
Angle Bisector Theorem

$\dfrac{AD}{AB} = \dfrac{CD}{BC}$

Never knew that this theorem exists. Gotta learn how it was derived from now.
 

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