In triangle ABC, angle B is a right angle. The length of side AB is 10√41 and the length of side BC is 24√41. What is the side length of AC?
A. 14√41
B. 26√41
C. 34√41
D. 34√41
Answer (2)
To find the length of side A C in triangle A BC , where ∠ B is a right angle, we will use the Pythagorean Theorem. The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In triangle A BC , since ∠ B is the right angle, A C is the hypotenuse. We know:
A B = 10 41
BC = 24 41
According to the Pythagorean Theorem:
A C 2 = A B 2 + B C 2
Substitute the given values:
A C 2 = ( 10 41 ) 2 + ( 24 41 ) 2
Calculate each term:
( 10 41 ) 2 = 100 × 41 = 4100
( 24 41 ) 2 = 576 × 41 = 23616
Add the values:
A C 2 = 4100 + 23616 = 27716
To find A C , take the square root of both sides:
A C = 27716
Notice that 27716 has been expressed in terms of 41 , implying that:
A C = 34 41
Thus, the length of side A C is 34 41 .
Therefore, the correct answer is option C : 34 41 .
Using the Pythagorean Theorem, we can determine that the length of side AC in triangle ABC equals 34√41. Thus, the correct option is C: 34√41.
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