1)In the figure, the vertices of right-angled isosceles triangle ABC are A, B(2,0) and C(8,0) respectively. find the equations of AB and AC
2)If the x-intercept and y-intercept of the straight line L: B(y-2) =Ax+8 are -8 and 3 respectively, find
a)the value of A and B
b)the slope of the straight line L.
1) To find the equations of AB and AC in the right-angled isosceles triangle ABC, we need to determine the slope of each line and their respective y-intercepts.
First, let's calculate the slope of AB. The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Here, the coordinates of A are (2, 0) and the coordinates of B are (8, 0). Plugging these values into the slope formula, we have:
slope_AB = (0 - 0) / (8 - 2)
= 0 / 6
= 0
Since the triangle is a right-angled isosceles triangle, AB is a horizontal line. For a horizontal line, the slope is always 0. So, the equation of AB is y = 0.
Next, let's calculate the slope of AC. Again, using the slope formula with the coordinates of A (2, 0) and C (8, 0):
slope_AC = (0 - 0) / (8 - 2)
= 0 / 6
= 0
Similarly to AB, AC is also a horizontal line with a slope of 0. Therefore, the equation of AC is also y = 0.
So, the equations of AB and AC are both y = 0.
2) Now let's solve the linear equation L: B(y-2) = Ax + 8 for the given x-intercept and y-intercept.
a) The x-intercept is the point where the line intersects the x-axis, which means that the y-coordinate is 0. So, we can substitute y = 0 into the equation:
B(0 - 2) = Ax + 8
-2B = Ax + 8
Since the x-intercept is -8, we know that when y = 0, x = -8. Plugging in these coordinates, we have:
-2B = A(-8) + 8
-2B = -8A + 8
The y-intercept is the point where the line intersects the y-axis, which means that the x-coordinate is 0. Substituting x = 0 into the equation:
B(y - 2) = A(0) + 8
B(y - 2) = 8
By - 2B = 8
By = 8 + 2B
We also know that the y-intercept is 3. Therefore, when x = 0, y = 3. Plugging these values:
3B = 8 + 2B
From the equation -2B = -8A + 8, we can isolate B:
-2B + 8A = 8
B = (8 - 8A) / -2
B = 4A - 4
Substituting this back into the equation 3B = 8 + 2B:
3(4A - 4) = 8 + 2(4A - 4)
12A - 12 = 8 + 8A - 8
12A - 12 = 8A
4A = 12
A = 3
Now we can substitute A = 3 into the equation B = 4A - 4:
B = 4(3) - 4
B = 12 - 4
B = 8
So, the value of A is 3 and the value of B is 8.
b) The slope of the straight line L can be determined using the formula:
slope = -A / B
Plugging in the values, we have:
slope = -3 / 8
Therefore, the slope of the straight line is -3/8.