SSC CGL Exam > SSC CGL Questions > In a triangle ABC, ∠BAC = 90o and AD is perp...
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In a triangle ABC, ∠BAC = 90o and AD is perpendicular to BC. If AD = 6 cm and BD = 4 cm, then the length of BC is
- a)10 cm
- b)12 cm
- c)13 cm
- d)15 cm
Correct answer is option 'B'. Can you explain this answer?
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In a triangle ABC, ∠BAC = 90o and AD is perpendicular to BC. If AD = ...
Solution:
Given: ∠BAC = 90°, AD ⊥ BC, AD = 6 cm, BD = 4 cm
Let's consider △ABD and △ADC.
In △ABD,
BD = 4 cm
AD = 6 cm
Using Pythagoras theorem,
AB² = AD² + BD²
AB² = 6² + 4²
AB² = 36 + 16
AB² = 52
AB = √52
AB = 2√13
In △ADC,
AD = 6 cm
AC = AB + BC
AC = 2√13 + BC
Using Pythagoras theorem,
AC² = AD² + DC²
(AB + BC)² = AD² + DC²
(2√13 + BC)² = 6² + DC²
4 × 13 + 4 × 2√13 × BC + BC² = 36 + DC²
52 + 8√13 × BC + BC² = 36 + DC²
Now, we need to find out the value of DC².
In △ABC,
AB² + BC² = AC²
(2√13)² + BC² = AC²
52 + BC² = AC²
Substitute the value of AC² in the above equation, we get
52 + BC² = 52 + 8√13 × BC + DC²
BC² - 8√13 × BC - DC² = 0
Using the quadratic formula,
BC = [8√13 ± √(64 × 13 + 4DC²)]/2
BC = 4√13 ± √(208 + DC²)
Since BC cannot be negative, we take the positive value
BC = 4√13 + √(208 + DC²)
Substitute the value of DC² in the above equation, we get
BC = 4√13 + √(208 + 36)
BC = 4√13 + √244
BC = 4√13 + 2√61
Now, we need to check which option is correct.
Option A: BC = 10 cm
Substitute the value of BC in the equation AB² + BC² = AC²
(2√13)² + (10)² = AC²
52 + 100 = AC²
AC = √152
AC = 2√38
Using Pythagoras theorem in △ADC,
AC² = AD² + DC²
(2√13 + 10)² = 6² + DC²
(2√13 + 10)² - 6² = DC²
(4√13 + 4) × (4√13 - 4) = DC²
192 = DC²
DC = √192
DC = 8√3
But, BC ≠ 10 cm. Hence option A is incorrect.
Option B: BC = 12 cm
Substitute the value of BC in the equation AB² + BC² = AC²
(2√13)² + (12)² = AC²
52 + 144 = AC²
AC
Given: ∠BAC = 90°, AD ⊥ BC, AD = 6 cm, BD = 4 cm
Let's consider △ABD and △ADC.
In △ABD,
BD = 4 cm
AD = 6 cm
Using Pythagoras theorem,
AB² = AD² + BD²
AB² = 6² + 4²
AB² = 36 + 16
AB² = 52
AB = √52
AB = 2√13
In △ADC,
AD = 6 cm
AC = AB + BC
AC = 2√13 + BC
Using Pythagoras theorem,
AC² = AD² + DC²
(AB + BC)² = AD² + DC²
(2√13 + BC)² = 6² + DC²
4 × 13 + 4 × 2√13 × BC + BC² = 36 + DC²
52 + 8√13 × BC + BC² = 36 + DC²
Now, we need to find out the value of DC².
In △ABC,
AB² + BC² = AC²
(2√13)² + BC² = AC²
52 + BC² = AC²
Substitute the value of AC² in the above equation, we get
52 + BC² = 52 + 8√13 × BC + DC²
BC² - 8√13 × BC - DC² = 0
Using the quadratic formula,
BC = [8√13 ± √(64 × 13 + 4DC²)]/2
BC = 4√13 ± √(208 + DC²)
Since BC cannot be negative, we take the positive value
BC = 4√13 + √(208 + DC²)
Substitute the value of DC² in the above equation, we get
BC = 4√13 + √(208 + 36)
BC = 4√13 + √244
BC = 4√13 + 2√61
Now, we need to check which option is correct.
Option A: BC = 10 cm
Substitute the value of BC in the equation AB² + BC² = AC²
(2√13)² + (10)² = AC²
52 + 100 = AC²
AC = √152
AC = 2√38
Using Pythagoras theorem in △ADC,
AC² = AD² + DC²
(2√13 + 10)² = 6² + DC²
(2√13 + 10)² - 6² = DC²
(4√13 + 4) × (4√13 - 4) = DC²
192 = DC²
DC = √192
DC = 8√3
But, BC ≠ 10 cm. Hence option A is incorrect.
Option B: BC = 12 cm
Substitute the value of BC in the equation AB² + BC² = AC²
(2√13)² + (12)² = AC²
52 + 144 = AC²
AC
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Community Answer
In a triangle ABC, ∠BAC = 90o and AD is perpendicular to BC. If AD = ...
The given data can be represented diagrammatically as follows.
Triangles ABC, ADB and ADC are right angled triangles.
By Pythagoras theorem,
BC2 = AB2 + AC2
⇒ (BD + DC)2 = (AD2 + DB2) + (AD2 + DC2)
⇒ (4 + DC)2 = 62 + 42 + 62 + DC2 (∵ (a + b)2 = a2 + 2ab + b2)
⇒ 16 + 8DC + DC2 = 36 + 16 + 36 + DC2
⇒ 8DC = 72
⇒ DC = 72/8 = 9 cm
BC = BD + DC = 4 + 9 = 13 cm
∴ BC = 13 cm
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In a triangle ABC, ∠BAC = 90o and AD is perpendicular to BC. If AD = 6 cm and BD = 4 cm, then the length of BC isa)10 cmb)12 cmc)13 cmd)15 cmCorrect answer is option 'B'. Can you explain this answer?
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