17.3 - Basic Analytic Geometry Problems

Links and useful resources

Concept summary and connections

Making connections between concepts from different representations: formula for midpoint yields same result as bisector, etc.

Lesson and worked examples

You know how to find the midpoint of a segment with compass and straightedge, and you know how to compute it from the coordinates of the end points. This lesson is all about connecting those two worlds so that you can use them together. Everything you did in straight geometry has a counterpart in analytic geometry, and all of the proof results and theorems are still true. Analytic geometry just gives you more tools to find those crucial points, segments, areas, etc.

So, as you approach these problems, it might help to think, "How would I do this with just compass and straightedge?" to get an idea of the sequence of steps you need. Then, you can think, "Now, how can I get the important points using what I know from the analytic description of the problem?"

Sticking the two ways of thinking together is not easy, but it's very powerful.

Another helpful thing you can do is to try to come up with all of the formulas you know that relate to the problem's different kinds of quantities (lengths, locations, areas, volumes) and then also think of the results from geometry that you know that could be related, and see how you might bounce back and forth between them to break through sticking points in the problem.

Guided practice

The area of the shaded region between line $k$ and the axes in the diagram at right is 36. If line $k$ passes through $(12, 0)$ , then what is the slope of $k$ ?
Let $A$ be $(- 5, 6)$ , $B$ be $(- 7, 9)$ , and $C$ be $(1, 10)$ .
- (a) Find the length of the median of $△ A B C$ from $B$ to $\overset{―}{A C}$ .
- (b) Find the slope of each side of $△ A B C$ . Notice anything interesting?
- (c) Find the area of $△ A B C$ .
- (d) Find the length of the altitude from $A$ to $\overset{―}{B C}$ .
In this problem, we determine the number of points $(a, b)$ such that $a^{2} + b^{2} = 25$ and the area of the triangle with vertices $(- 5, 0)$ , $(5, 0)$ , and $(a, b)$ is 10.
- (a) What does the fact that $(a, b)$ satisfies $a^{2} + b^{2} = 25$ tell us about where the point $(a, b)$ is located on the coordinate plane?
- (b) What does the fact that the area of the triangle with vertices $(- 5, 0)$ , $(5, 0)$ and $(a, b)$ is 10 tell us about where the point $(a, b)$ is located on the coordinate plane?
- (c) How many points satisfy the conditions you found in parts (a) and (b)?
$A B C D$ is a rectangle with $A B = 8$ and $A D = 12$ . Points $X$ and $Y$ are on $\overset{―}{A B}$ and $\overset{―}{C D}$ , respectively, such that $A X = 7 / 3$ and $D Y = 1$ . $\overset{―}{X Y}$ meets diagonal $\overset{―}{A C}$ at point $P$ . In this problem, we find $B P$ .
- (a) Set the problem up on the Cartesian plane. Let $A$ be the origin. What coordinates can we assign to each of $B$ , $C$ , and $D$ ?
- (b) What are the coordinates of $P$ ?
- (c) Find $B P$ .

Homework

#hw (geom-gr10) 17.3 Basic Analytic Geometry Problems (p. 446) (gr10) 2025-05-02