How do I find the answer to the following question without knowing the 3rd vertices?
Graph a triangle with vertices (3,-2), (-3,2) and height of 3 units.
The answer is supposed to be (3, -2), (0,1) and (-3, -2).
If your answer is correct, you made a typo in the 1st point of the problem.
If two of the vertices are
(3,-2) and (-3,-2) then the base is clearly a horizontal line.
So, any point 3 units above the base will produce the desired height. That would be (x,1).
(0,1) is the point that makes the triangle isosceles, since it's midway between the two base vertices.
Getting the right answer is a lot easier if the right questions is asked. If the question was posed as you entered it, it was a very poor example.
and vertex is the singular of vertices!
Yes, that was how the question was asked. Not sure how I'm supposed to take your response about using wrong spelling.
To find the missing vertex of the triangle, you can use the given information about the height of the triangle.
First, let's understand what the height of a triangle means. The height of a triangle is the perpendicular distance between the base of the triangle and its opposite vertex. In this case, the base of the triangle can be considered as the line segment connecting the given vertices (3,-2) and (-3,2).
To find the missing vertex, we need to determine the equation of the line that represents the base of the triangle.
Step 1: Determine the slope of the line connecting the given vertices.
- Let's denote the first given vertex as A(3, -2) and the second given vertex as B(-3, 2).
- The slope of the line passing through two points can be calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points.
- Calculating the slope of line AB: m = (2 - (-2)) / (-3 - 3) = 4 / (-6) = -2/3.
Step 2: Find the equation of the line using point-slope form.
- Using point-slope form, y - y1 = m(x - x1), where (x1, y1) is a known point on the line and m is the slope we calculated in step 1.
- We can use either vertex A or B as the known point.
- Let's use vertex A(3, -2).
- The equation of the line passing through A(3, -2) with a slope of -2/3 is: y - (-2) = (-2/3)(x - 3).
- Simplifying the equation: y + 2 = (-2/3)x + 2.
- Rearranging the equation: y = (-2/3)x.
Step 3: Determine the equation of the line that represents the height.
- We are given that the height of the triangle is 3 units.
- Since the height is perpendicular to the base, its slope will be the negative reciprocal of the base's slope.
- The base's slope is -2/3, so the height's slope will be 3/2 (flipped and with opposite sign).
- We need to determine the equation of the line passing through the mid-point of the base (since the height is from the base to the opposite vertex) and with a slope of 3/2.
- The mid-point of the base can be calculated as follows:
- x-coordinate: (x1 + x2) / 2 = (3 + (-3)) / 2 = 0 / 2 = 0.
- y-coordinate: (y1 + y2) / 2 = (-2 + 2) / 2 = 0 / 2 = 0.
- Therefore, the midpoint is (0, 0).
- Let's denote the midpoint as M(0, 0).
- The equation of the line passing through M(0, 0) with a slope of 3/2 is: y - 0 = (3/2)(x - 0).
- Simplifying the equation: y = (3/2)x.
Step 4: Find the intersection point of the two lines.
- Now, we have the equations of the base line (y = (-2/3)x) and the height line (y = (3/2)x).
- To find the intersection point, we need to solve these two equations simultaneously.
- Equating the two equations: (-2/3)x = (3/2)x.
- Multiplying both sides of the equation by 6 (the least common multiple of the denominators): -4x = 9x.
- Adding 4x to both sides: 9x + 4x = 0.
- Simplifying the equation: 13x = 0.
- Dividing both sides by 13: x = 0.
- Substituting the value of x into any of the two equations, we get: y = (3/2)(0) = 0.
- Therefore, the intersection point is (0, 0).
So, the missing vertex of the triangle is (0, 0). Now, we have all three vertices of the triangle: A(3, -2), B(-3, 2), and C(0, 0).