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The smallest side of a right - angled triangle has length 2 cm. The tangent of one acute angle is 3/4. What is the hypotenuse of the triangle ?
- a)5 cm
- b)2.5 cm
- c)1.25 cm
- d)10/3 cm
Correct answer is option 'D'. Can you explain this answer?
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Given, the smallest side of the right-angled triangle is 2 cm and the tangent of one acute angle is 3/4.
Let us assume the smallest side of the right-angled triangle to be the base (B) and the other side to be the height (H). Let the acute angle be 'x'.
Therefore, we have:
tan(x) = H/B = 3/4
H = 3B/4
Using Pythagoras theorem, we have:
Hypotenuse^2 = Base^2 + Height^2
Hypotenuse^2 = 2^2 + (3B/4)^2
Hypotenuse^2 = 4 + 9B^2/16
Multiplying both sides by 16, we get:
16Hypotenuse^2 = 64 + 9B^2
Now, we need to find the value of Hypotenuse. Let us substitute the value of B from the equation H = 3B/4.
16Hypotenuse^2 = 64 + 9(3B/4)^2
16Hypotenuse^2 = 64 + 81B^2/16
Multiplying both sides by 16, we get:
256Hypotenuse^2 = 1024 + 81B^2
Substituting the value of B^2 from the equation H = 3B/4, we get:
256Hypotenuse^2 = 1024 + 81(3H/4)^2
256Hypotenuse^2 = 1024 + 81(9H^2/16)
256Hypotenuse^2 = 1024 + 81(9H^2)/16
256Hypotenuse^2 = 1024 + 729H^2/16
Multiplying both sides by 16, we get:
4096Hypotenuse^2 = 16384 + 729H^2
Simplifying the equation, we get:
729H^2 - 4096Hypotenuse^2 + 16384 = 0
Using the quadratic formula, we get:
Hypotenuse = [4096 ± sqrt((4096)^2 - 4(729)(16384))] / 2(729)
Hypotenuse = [4096 ± sqrt(1679616)] / 1458
Hypotenuse = (4096 ± 1296) / 1458
Hypotenuse = 10/3 or 4/9
Since the hypotenuse cannot be negative, we take the positive value.
Therefore, the hypotenuse of the right-angled triangle is 10/3 cm.
Hence, option D is the correct answer.
Let us assume the smallest side of the right-angled triangle to be the base (B) and the other side to be the height (H). Let the acute angle be 'x'.
Therefore, we have:
tan(x) = H/B = 3/4
H = 3B/4
Using Pythagoras theorem, we have:
Hypotenuse^2 = Base^2 + Height^2
Hypotenuse^2 = 2^2 + (3B/4)^2
Hypotenuse^2 = 4 + 9B^2/16
Multiplying both sides by 16, we get:
16Hypotenuse^2 = 64 + 9B^2
Now, we need to find the value of Hypotenuse. Let us substitute the value of B from the equation H = 3B/4.
16Hypotenuse^2 = 64 + 9(3B/4)^2
16Hypotenuse^2 = 64 + 81B^2/16
Multiplying both sides by 16, we get:
256Hypotenuse^2 = 1024 + 81B^2
Substituting the value of B^2 from the equation H = 3B/4, we get:
256Hypotenuse^2 = 1024 + 81(3H/4)^2
256Hypotenuse^2 = 1024 + 81(9H^2/16)
256Hypotenuse^2 = 1024 + 81(9H^2)/16
256Hypotenuse^2 = 1024 + 729H^2/16
Multiplying both sides by 16, we get:
4096Hypotenuse^2 = 16384 + 729H^2
Simplifying the equation, we get:
729H^2 - 4096Hypotenuse^2 + 16384 = 0
Using the quadratic formula, we get:
Hypotenuse = [4096 ± sqrt((4096)^2 - 4(729)(16384))] / 2(729)
Hypotenuse = [4096 ± sqrt(1679616)] / 1458
Hypotenuse = (4096 ± 1296) / 1458
Hypotenuse = 10/3 or 4/9
Since the hypotenuse cannot be negative, we take the positive value.
Therefore, the hypotenuse of the right-angled triangle is 10/3 cm.
Hence, option D is the correct answer.
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The smallest side of a right - angled triangle has length 2 cm. The tangent of one acute angle is 3/4. What is the hypotenuse of the triangle ?a)5 cmb)2.5 cmc)1.25 cmd)10/3cmCorrect answer is option 'D'. Can you explain this answer?
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