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Two vertices of a triangle are (0,2) and (4,3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?
- a)Fourth
- b)Second
- c)Third
- d)First
Correct answer is option 'B'. Can you explain this answer?
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Two vertices of a triangle are (0,2) and (4,3).If its orthocentre is a...
⇒ 3b – 6 = –4a ⇒ 4a + 3b = 6 ……(ii)
From (i) and (ii)
∴ IInd quadrant.
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Two vertices of a triangle are (0,2) and (4,3).If its orthocentre is a...
To find the third vertex of the triangle, we need to consider the property of the orthocenter. The orthocenter is the point of intersection of the altitudes of a triangle. Since the orthocenter is at the origin, it means that the altitudes of the triangle pass through the origin.
Let's label the given vertices of the triangle as A(0, 2) and B(4, 3). The third vertex will be labeled as C(x, y).
- Let's find the slope of the line AB using the formula: m = (y2 - y1) / (x2 - x1)
m = (3 - 2) / (4 - 0) = 1/4
- Since the slope of AB is 1/4, the slope of the altitude drawn from C to AB will be the negative reciprocal of 1/4. Let's call this slope m1.
m1 = -1 / (1/4) = -4
- We know that the line passing through the origin (0, 0) and having a slope of -4 is given by the equation y = -4x.
- Substituting the coordinates of C(x, y) into the equation y = -4x, we get y = -4x.
- Now let's substitute the x-coordinate of B into the equation y = -4x to find the y-coordinate of C.
3 = -4(4)
3 = -16
- Since the equation -16 = 3 is not satisfied, it means that there is no solution for C(x, y) in the second quadrant.
- Let's check for the other quadrants.
- For the first quadrant, x > 0 and y > 0. But the equation y = -4x does not satisfy this condition.
- For the third quadrant, x < 0="" and="" y="" />< 0.="" but="" the="" equation="" y="-4x" does="" not="" satisfy="" this="" />
- Therefore, the third vertex C(x, y) can only lie in the fourth quadrant.
Hence, the correct answer is option B) Fourth quadrant.
Let's label the given vertices of the triangle as A(0, 2) and B(4, 3). The third vertex will be labeled as C(x, y).
- Let's find the slope of the line AB using the formula: m = (y2 - y1) / (x2 - x1)
m = (3 - 2) / (4 - 0) = 1/4
- Since the slope of AB is 1/4, the slope of the altitude drawn from C to AB will be the negative reciprocal of 1/4. Let's call this slope m1.
m1 = -1 / (1/4) = -4
- We know that the line passing through the origin (0, 0) and having a slope of -4 is given by the equation y = -4x.
- Substituting the coordinates of C(x, y) into the equation y = -4x, we get y = -4x.
- Now let's substitute the x-coordinate of B into the equation y = -4x to find the y-coordinate of C.
3 = -4(4)
3 = -16
- Since the equation -16 = 3 is not satisfied, it means that there is no solution for C(x, y) in the second quadrant.
- Let's check for the other quadrants.
- For the first quadrant, x > 0 and y > 0. But the equation y = -4x does not satisfy this condition.
- For the third quadrant, x < 0="" and="" y="" />< 0.="" but="" the="" equation="" y="-4x" does="" not="" satisfy="" this="" />
- Therefore, the third vertex C(x, y) can only lie in the fourth quadrant.
Hence, the correct answer is option B) Fourth quadrant.
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Two vertices of a triangle are (0,2) and (4,3).If its orthocentre is at the origin, then its third vertex lies in which quadrant?a)Fourthb)Secondc)Thirdd)FirstCorrect answer is option 'B'. Can you explain this answer?
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