In the given figure, AD = 13 cm, BC = 12 cm, AB = 3 cm and angle ACD = angle ABC = 90°. Find the length of DC.
Understand the Problem
The question involves finding the length of certain sides in right triangles using given dimensions and angles. Specifically, it includes calculating lengths within right triangles that are derived from the given measurements and relationships between the angles.
Answer
The length of \( BC \) is \( 23 \, \text{cm} \).
Answer for screen readers
The length of ( BC ) is ( 23 , \text{cm} ).
Steps to Solve
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Identify the right triangle components In triangle ADC, we have ( AD = 10 , \text{cm} ), ( DC = 24 , \text{cm} ), and we need to find ( AC ). Since ( \angle ADB = 90^\circ ), we can apply the Pythagorean theorem.
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Apply the Pythagorean Theorem Using the theorem, we have: $$ AC^2 = AD^2 + DC^2 $$ Substituting the given values: $$ AC^2 = 10^2 + 24^2 $$
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Calculate the squares Calculate each term: $$ AC^2 = 100 + 576 $$ Then sum them: $$ AC^2 = 676 $$
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Find the length of AC Take the square root to find ( AC ): $$ AC = \sqrt{676} = 26 , \text{cm} $$
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Repeat for segment CD Next, note that triangle ADB is also a right triangle. Since ( AC = AB + BC ) and we found ( AC = 26 , \text{cm} ), and given ( AB = 3 , \text{cm} ), we have: $$ BC = AC - AB $$ Substituting the values: $$ BC = 26 - 3 = 23 , \text{cm} $$
The length of ( BC ) is ( 23 , \text{cm} ).
More Information
In this right triangle problem, we used the Pythagorean theorem to find the lengths of sides in relationships defined by the angles. This is a common approach in trigonometry.
Tips
- Misapplying the Pythagorean theorem to non-right triangles.
- Forgetting to take the square root when finding lengths.
- Not using consistent units, which could lead to errors.
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