什么是平行线公式？
两条或多条无限延伸且永不相交的线的组合称为平行线或共面线。平行线用特殊符号表示，由 || 给出。
特性：
- 如果两条线之间保持恒定的垂直距离，则称两条线平行
- 两条平行线的斜率相等
- 他们从不相交
- 它们始终与任何轴具有相等的底角。
如果线 a 和 b 平行，则∠A = ∠B
与平行线相关的角度
- 交替角：它们是当一条线与两条线相交时形成的角度，它们位于横向线的相对侧和其他线的相对侧。如果两条线相等，则交替的角度也相等。
这里，交替的角度相等，∠A = ∠B
- 对应角：横线与任意一对线相交时所形成的角称为对应角。如果两条线平行，则对应的角度也相等。
这里，对应的角度相等，∠A = ∠B
- 内角：如果一条导线与两条平行线相交，则该导线同一侧的每一对内角之和是互补的。
∠A + ∠B = 180
线的斜率
斜率是确定范围线是否指向轴的测量值。通常，我们将斜率定义为 X 轴与直线之间的角度的切线。如果 X 轴与给定直线的夹角为 θ，则
这里，tanθ 是斜率，b 是截距
Slope = tanθ
1.对于方程y = mx + b的直线，我们定义：
- m：给定线的斜率
- b： Y轴上的直线截距。
2.如果给定直线上的两个点(a,b)和(c,d) ，那么我们可以使用以下公式求直线的斜率：
(a,b) 和 (c,d) 是线上的两个点
Slope = (b – d) / (a – c)
可以通过考虑线上的点 (x, y) 来确定线的方程。因为我们也可以根据 x 和 y 找到斜率，所以我们可以写
斜率 = (y – d) / (x – c)
将其与斜率的实际值相等，我们可以推导出给定线的等式，
(y – d) / (x – c) = (b – d) / (a – c)
3.对于给定的直线ax + by + c = 0 ，我们可以将斜率写为：
Slope = – a / b
示例问题
问题 1. 线 y = 2x + 3 和 5y = 10x + 21 平行吗？
解决方案：
Given the lines are
y = 2x + 3 — (i)
5y = 10x + 21 — (ii)
Dividing equation (ii) with 5, we get:
5y/5 = 10x/5 + 21/5
y = 2x + 4.2
Thus, we see that the slope of both the lines are equal to 2, hence the lines are parallel.
问题 2. 确定直线 2x + 5y + 6 = 0 的斜率。
解决方案：
Since, the given line is in the form of ax + by + c = 0, we can compare the values of a , b and c and get their values as,
a = 2, b = 5 and c = 6
Thus, the slope of the line = -a/b = – 2/5 = -0.40
问题 3：确定通过 (1, 2) 并平行于线 y = 2x + 5 的线的方程。
解决方案：
The given line y = 2x + 5 has slope equals to 2.
We consider any point (x, y) on the line of which we need to find the equation. So, we can write the slope of the line as:
Slope = (y – 2)/(x – 1)
Since this line is parallel to y = 2x + 5, ao the slope of the line is also equal to 2. So, we can write:
(y – 2) / (x – 1) = 2
y – 2 = 2x – 2
y = 2x
So, the equation of the line is y = 2x
问题 4. 如果一条线通过点 (k, 2) 和 (6, 7) 并平行于线 3x – 3y + 5 = 0，则求 k 的值。
解决方案：
Given line is parallel to the line 3x – 3y + 5 = 0. Since, this is in the form of ax + by + c = 0, we can write,
Slope of this line = – a / b = – 3 / (-3) = 1
In terms of the two points, we can write the slope of the given line as,
Slope = (7 – 2) / (6 – k) = 5 / (6 – k)
Since, the lines are parallel, we can write:
5 / (6 – k) = 1
6 – k = 5
k = 1
Hence, the value of k is 1.
问题 5. 一条直线平行于直线 y = x + 3. 求直线与 X 轴之间的内角。
解决方案：
Since the lines are parallel, they must have the same slope.
So, the slope of the given line = Slope of y = x + 3 = 1
Lets, the angle made by the line and X-axis be θ, then we can write:
tanθ = 1
θ = tan-1(1) =45o
So, the angle between the line and X-axis is equal to 45o.