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

In summary, using the Angle Bisector Theorem, we can find the length of AD, which is the bisector of angle B in a right triangle ABC with AB = 12 cm and BC = 15 cm. By solving the equation $\dfrac{AD}{12} = \dfrac{9-AD}{15}$, we get AD = 4 cm. This theorem is important in understanding the length ratio of triangles with special angles, such as 45°-90°-45° or 30°-60°-90°, and can be derived from the Pythagorean theorem.
  • #1
Monoxdifly
MHB
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0
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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  • #2
Pythagoras $\implies AC = 9$

using the ...

Angle Bisector Theorem

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

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

solve for $AD$
 
  • #3
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.
 

1. How do you find the length of the hypotenuse in a right triangle with a 45° angle?

The length of the hypotenuse in a right triangle with a 45° angle can be found by using the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, since the triangle has two sides with equal length, the formula simplifies to c = √2 * a, where c is the length of the hypotenuse and a is the length of one of the other sides.

2. How do you find the length of the other two sides in a right triangle with a 45° angle?

The length of the other two sides in a right triangle with a 45° angle can be found by using the fact that all three angles in a triangle add up to 180°. Since we know one angle is 45° and the right angle is 90°, the third angle must be 45° as well. This means that the triangle is an isosceles right triangle, and the two other sides must have equal length. Therefore, the length of each of the other two sides can be found by dividing the length of the hypotenuse by √2.

3. What is the relationship between the sides in a right triangle with a 45° angle?

In a right triangle with a 45° angle, the relationship between the sides is that the two shorter sides are equal in length and the length of the hypotenuse is equal to the length of one of the shorter sides multiplied by √2. This is because the triangle is an isosceles right triangle, and the Pythagorean theorem can be used to find the length of the hypotenuse.

4. Can the Pythagorean theorem be used to find the length of the sides in a right triangle with a 45° angle?

Yes, the Pythagorean theorem can be used to find the length of the sides in a right triangle with a 45° angle. This theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In the case of a right triangle with a 45° angle, the formula simplifies to c = √2 * a, where c is the length of the hypotenuse and a is the length of one of the other sides.

5. How can the angles in a right triangle with a 45° angle be used to find the length of the sides?

The angles in a right triangle with a 45° angle can be used to find the length of the sides by using the fact that all three angles in a triangle add up to 180°. Since we know one angle is 45° and the right angle is 90°, the third angle must be 45° as well. This means that the triangle is an isosceles right triangle, and the two other sides must have equal length. Therefore, the length of each of the other two sides can be found by dividing the length of the hypotenuse by √2.

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