find the equation of the straight line through A(3,20) which is perpendicular to the line passing through the points B(-2,8) and C(11,8). Hence, find the area of the triangle ABC, giving your answer in the simplest possible form.
slope of BC = (8-8)/(11=2) = 0
so BC is a horizontal line
Thus the line through A must be a vertical line
All vertical lines have the form x = something
Look at the given point A, what is its x value ?
so the equation of your required line is
x = 3
Make a sketch.
The base BC of the triangle is 13 units long, and A is 12 units above BC
so the base is 13 and the height is 12
Area = (1/2)(13)(12) = 78 units^2
Why is the base BC is 13, its because x2-x1? and why is 12 in A?
BC = 11 - (-2) = 13
what is the difference between a height of 8 and a height of 20 ??
Did you look at your diagram?
oh, okay. thank you very much for your help
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the original line. Then we can use this slope to find the slope of the perpendicular line.
1. Determine the slope of the line passing through points B(-2,8) and C(11,8).
- The formula for slope (m) is: m = (y2 - y1) / (x2 - x1)
- Substituting the given points, we get: m = (8 - 8) / (11 - (-2))
- Simplifying, we find that the slope is 0.
2. The slope of the perpendicular line will be the negative reciprocal of the slope of the original line.
- Since the slope of the original line is 0, the slope of the perpendicular line will be undefined (as it is with vertical lines).
3. To find the equation of the perpendicular line passing through point A(3,20), we know that the line is vertical (parallel to the y-axis) and intersects the x-coordinate of 3.
- Therefore, the equation for this line will simply be x = 3. (Since x can take any value and y can be any real number.)
Now, let's move on to calculating the area of triangle ABC.
4. We have points A(3,20), B(-2,8), and C(11,8).
- To find the area of a triangle, we use the formula: Area = 0.5 * base * height.
5. The base (b) of the triangle is the distance between points A and B.
- Using the distance formula: b = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Substituting the coordinates: b = sqrt((-2 - 3)^2 + (8 - 20)^2)
- Simplifying, we find that the base is 7√10.
6. The height (h) of the triangle is the perpendicular distance from point C to line BA.
- Since line BA is vertical (parallel to the y-axis), the height is simply the difference in y-coordinates between C and any point on line BA.
- The height will be 8 - 20 = -12.
7. Now, calculate the area using the formula: Area = 0.5 * base * height.
- Substituting the values: Area = 0.5 * 7√10 * (-12)
- Simplifying, we get: Area = -42√10.
Thus, the equation of the perpendicular line is x = 3, and the area of triangle ABC is -42√10.