中三數學問題求幫忙(急)

1. A straight line with slope 1.
Slope = rise/run = 1
Inclination = tan^1 (1) = 45°

2. A straight line passing through A(1,-3)and B(-5,-7).
A(x1, y1), B(x2, y2)
slope = (y2 – y1)/(x2 – x1) =[ -7 – (-3)]/[-5 -1] = -4/-6 = 2/3
Inclination = tan^1 (2/3) = 33.7°

3. A(8,-2)and B(14,8)
A(x1, y1), B(x2, y2)
midpoint of AB= (x1+x2)/2, (y1 + y2)/2)
= (8+14)/2, (-2 + 8)/2)
midpoint of AB (11, 3)

4. P(1.5,12)and Q(-7.5,-2)
P(x1, y1), Q(x2, y2)
midpoint of PQ = (x1+x2)/2, (y1 + y2)/2)
= (1.5-7.5)/2, (12 -2)/2)
midpoint of PQ (-3, 5)

5. Consider the 3 points A(-3,2),B(4,4)and C(11,2). Prove that triangle ABC is an
isosceles triangle.
If A(x1, y1), B(x2, y2),
distance between A and B =square root of [(x2-x1)^2 + (y2 – y1)^2]

distance between A and B =square root of [(4 + 3)^2 + (4 – 2)^2] = SqR (53)
distance between B and C =square root of [(11 -4)^2 + (2 – 4)^2] = SqR (53)
distance between C and A =square root of [(-3 -11)^2 + (2 – 2)^2] = 14

Since AB = BC = SqR(53) , triangle ABC is an isosceles triangle.
Isosceles triangle has two equal sides

Note: SqR = square root

6.Consider the 4 points A(10,9),B(8,3),C(1,1)and D(3,7). Prove that ABCD is a
parallelogram.
A(x1, y1), B(x2, y2)
slope AB = (3 – 9)/(8 – 10) = (-6)/ (-2) = 3
Slope CD = (7 – 1) /(3-1) =6/2 = 3
Slope BC = (1 – 3)/(1 – 8) = -2/-7 = 2/7
Slope DA = (9 - 7)/( 10-3) = 2/7

Since AB and CD have the same slope = 3, they are parallel
Since BC and DA have the same slope = 2/7, they are parallel
If opposite sides are parallel, ABCD is a parallelogram.