Question 597115: Maggie moved into a new house and was planning how to design the yard. Sitting on the front steps one day, she visualized a triangular flower bed, with one side along the right-hand edge of the straight front walkway from the steps to the road (as viewed from her vantage point on the steps).
The edge of the flowerbed closest to the house would start 2 feet from the steps, and would slope at a rate of 1 foot toward the street for every 4 feet to the right. The far edge of the triangle would begin at the edge of the walkway 9 feet from the steps. This side would slope at a rate of 5 feet toward the house for every 8 feet to the right.
Help Maggie draw up her flowerbed plans by representing the right-hand edge of the straight front walkway by the y-axis, the line containing the bottom edge of the bottom step as the x-axis, and the corner where the edge of the sidewalk meets the steps just below Maggie’s right foot as she sits on the steps by the origin. This places the region where she wants the flowerbed in the first quadrant. Maggie wants to plant a dogwood tree at the corner of this triangular flowerbed where the two angled sides meet.
Find an equation for each of the two angled sides
Find the coordinates of their point of intersection
Graph the region
Find the area of this triangular flowerbed
Show the position of the dogwood tree on your graph
Green line: AWAY from the house 1 for every 4 right:
Using the point-slope form:
Set the two left-hand sides equal to solve for the -coordinate of the point of intersection:
Solve for then substitute the value for back into either equation to find the -coordinate of the point of intersection. Examine the graph to make sure your answer makes sense.
Use the segment with the endpoints on the -axis as the base of the triangle. The measure of the base will be the difference between the -coordinates of the endpoints. Construct a perpendicular from the base through the point of intersection you calculated above. The segment from the base to the point of intersection will be the altitude and the measure of the altitude will be the -coordinate of the point of intersection.
The area of a triangle is the base times the height divided by 2.
My calculator said it, I believe it, that settles it