In the given figure, angle ADB = 90°, AC = 26 cm, AB = 26 cm, and AD = 24 cm; find the length of BC.
Understand the Problem
The question is asking to find the length of BC in a triangle given specific angles and side lengths. We have to use the properties of triangles and potentially the Pythagorean theorem since one angle is a right angle.
Answer
The length of $BC$ is $20 \, cm$.
Answer for screen readers
The length of $BC$ is $20 , cm$.
Steps to Solve
- Identify the Triangles
Since angle ADB is a right angle, we have a right triangle ABD. We will use triangle ABD to find the length of BD and then apply it to find BC.
- Apply the Pythagorean Theorem
In triangle ABD, we have:
- $AB = 26 , cm$ (hypotenuse)
- $AD = 24 , cm$ (one leg)
We can find the length of BD using the Pythagorean theorem, which states:
$$ AB^2 = AD^2 + BD^2 $$
Substituting in the known values:
$$ 26^2 = 24^2 + BD^2 $$
- Calculate BD
Calculate the squares:
$$ 676 = 576 + BD^2 $$
Now subtract $576$ from both sides:
$$ BD^2 = 676 - 576 = 100 $$
Taking the square root gives:
$$ BD = \sqrt{100} = 10 , cm $$
- Use the property of BD and DC
Given that $BD = DC$, we can conclude:
$$ DC = 10 , cm $$
- Find the length of BC
Now, we know:
- $BC = BD + DC = 10 , cm + 10 , cm = 20 , cm$
The length of $BC$ is $20 , cm$.
More Information
In this problem, we utilized the Pythagorean theorem to find the leg of the right triangle, which helped us determine the total length of segment $BC$. This approach is commonly used in problems involving right triangles, showing the interconnectedness of triangle properties.
Tips
- Incorrectly Applying the Pythagorean Theorem: Make sure to always match the correct sides when applying the theorem; the hypotenuse must be opposite the right angle.
- Forgetting to Apply the Lengths of Segments Properly: Ensure that both segments (BD) and (DC) are added accurately.
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