1. A,B and C are the points (-2,-1)(3,1)(2,-1) respectively. Find the point at which the line through C perpendicular to AB cuts the y-axis.
2. The vertices of triangle ABC are A(-2,2), B(1,1) and C (3,4). Find the slopes of the altitudes of the triangle.
3. The vertices of a quadrilateral ABCD are A(-4,-2), B(3,-1), C(4,4) and D (0,3). Show that its diagonals, i.e. AC and BD, are perpendicular to each other.
4.Show that the points P(-2,1), Q(-1,-3),R(3,-2) and S(2,2) form a square.
5. P,Q,R,S are the points (2,0),(4,-1),(6,3) and (4,4) respectively.
(a) Find the slopes of the sides of PQRS.
(b)What type of quadrilateral is PQRS?
F.3 maths(20分) 超urgent!
2008-11-04 1:52 am
回答 (1)
2008-11-04 4:12 am
✔ 最佳答案
1. The required line and AB are perpendicular to each other:
(Slope of the required line) x (Slope of AB) = -1
[(y+1)/(0-2)] x [(-1-1)/(-2-3)] = -1
[(y+1)/(-2)] x (2/5) = -1
(2y+2)/(-10) = -1
2y + 2 = 10
2y = 8
y = 4
Ans: The required point is (0, 4).
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2.
Slope of AB = (2-1)/(-2-1) = -1/3
Slope of the altitude from C to AB = -1/(-1/3) = 3
Slope of BC = (1-4)/(1-3) = 3/2
Slope of the altitude from A to BC = -1/(3/2) = -2/3
Slope of AC = (2-4)/(-2-3) = 2/5
Slope of the altitude from B to AC = -1/(2/5) = -5/2
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3.
Slope of AC = (-2-4)/(-4-4) = 3/4
Slope of BD = (-1-3)/(3-0) = -4/3
(Slope of AC) x (Slope of BD) = (3/4) x (-4/3) = -1
AC and BD are perpendicular to each other.
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4.
PQ = √[(-2+1)2 + (1+3)2] = √17
QR = √[(-1-3)2 + (-3+2)2] = √17
RS = √[(3-2)2 + (-2-2)2] = √17
SP = √[(2+2)2 + (2-1)2] = √17
Slope of PQ = (1+3)/(-2+1) = -4
Slope of QR = (-3+2)/(-1-3) = 1/4
(Slope of PQ) x (Slope of QR) = (-4)(1/4) = -1
PQR = 90o
Since PQ = QR = RS = SP and PQR is 90o, PQRS is a square.
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5.
(a)
Slope of PQ = (0+1)/(2-4) = -1/2
Slope of QR = (-1-3)/(4-6) = 2
Slope of RS = (3-4)/(6-4) = -1/2
Slope of SP = (4-0)/(4-2) = 2
(b)
PQ = √[(2-4)2 + (0+1)2] = √5
QR = √[(4-6)2 + (-1-3)2] = √20
Product of the slope of two adjacent sides = (-1/2) x (2) = -1
Hence, all 4 interior angles are 90o.
But adjacent sides are not equal.
Therefore, PQRS is a rectangle.
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