ISEE Upper Level

Coordinate Geometry

Plotting points, distance, midpoint, and slope on the coordinate plane — excludes graphing linear equations (see linear-graphing)

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Imagine a giant treasure map 🗺️ or a fun game of Battleship. That's exactly what the coordinate plane is! It is a grid made of two main lines: the x-axis (which goes left to right, like running across a soccer field) and the y-axis (which goes up and down, like climbing a tree). Every single point on this map has its own special address, written like (x, y).

When you plot a point, remember the golden rule: you always have to walk across the field (x) before you can climb the tree (y)! So for the point (3, 4), you walk 3 spaces right, and climb 4 spaces up.

On the ISEE, you get to be an explorer. You might need to find the distance between two spots (like your house and the pizza shop 🍕), find the exact middle point to meet a friend (that's called the midpoint), or figure out how steep a hill is (that's called slope).

Don't worry, you don't need to memorize a million confusing rules for the ISEE Math Achievement or Quantitative Reasoning sections. Coordinate geometry is really just about counting spaces on a grid and making simple triangles. Let's dive in and find some treasure! ⭐

Key Formula

To find the slope (how steep a line is):

Slope = \frac{change in y}{change in x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Practice Questions

3 practice questions for ISEE Upper Level

Q1 Hard

The endpoints of the diameter of a circle are

(- 4, 2)

and

(4, - 4)

. What is the radius of the circle?

A 5 grid units

B 6 grid units

C 8 grid units

D 10 grid units

Show Solution

First, find the length of the diameter by calculating the distance between the endpoints $(- 4, 2)$ and $(4, - 4)$ . The distance formula is $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ . Plugging in the points gives $(4 - (- 4))^{2} + (- 4 - 2)^{2} = 8^{2} + (- 6)^{2} = 64 + 36 = 100 = 10$ . The diameter is 10 grid units. The radius is half of the diameter, so $\frac{10}{2} = 5$ grid units.

Answer: A

Q2 Hard

Three vertices of a rectangle on a coordinate plane are located at

(- 2, 3)

, (4, 3), and

(4, - 5)

. What are the coordinates of the fourth vertex of the rectangle?

(- 2, - 5)

(- 5, - 2)

(4, - 2)

(- 2, 5)

Show Solution

A rectangle on a coordinate plane with horizontal and vertical sides shares x-coordinates and y-coordinates among its vertices. The given x-coordinates are $- 2$ and $4$ . The given y-coordinates are $3$ and $- 5$ . To complete the rectangle, the fourth vertex must use the remaining combination of these coordinates, which is an x-coordinate of $- 2$ and a y-coordinate of $- 5$ . Therefore, the fourth vertex is $(- 2, - 5)$ .

Answer: A

Q3 Hard

Point

M

is the midpoint of line segment

A B

. If the coordinates of

A

are

(- 3, 7)

and the coordinates of

M

are (1, 2), what are the coordinates of point

B

(- 1, 4.5)

(- 7, 12)

(4, - 5)

(5, - 3)

Show Solution

The midpoint formula is $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$ . Let the coordinates of point $B$ be (x, y). Setting up the equations for the x- and y-coordinates gives $\frac{- 3 + x}{2} = 1$ and $\frac{7 + y}{2} = 2$ . Solving for $x$ : $- 3 + x = 2$ , so $x = 5$ . Solving for $y$ : $7 + y = 4$ , so $y = - 3$ . The coordinates of point $B$ are $(5, - 3)$ .

Answer: D

Tips & Strategies

Always remember 'Run before you Jump!' When plotting a point (x, y), go left or right on the x-axis first, then go up or down on the y-axis.
For distance questions, don't panic if you forget the big distance formula. Just sketch a quick right triangle on your scratch paper and use the Pythagorean theorem ( $a^{2} + b^{2} = c^{2}$ ).
In Quantitative Comparison questions, if you're comparing slopes, quickly sketch the points! Sometimes you can literally see which line is steeper without doing any math at all.

Common Mistakes

Watch out for swapping your x and y coordinates! Writing (y, x) instead of (x, y) will put you in a completely different spot on the map.
Don't forget that slope is 'rise over run' ( $\frac{change in y}{change in x}$ ). A common mistake is flipping it upside down and doing run over rise!

Frequently Asked Questions

Do I need to memorize the huge distance formula for the ISEE?

Not really! The distance formula is actually just the Pythagorean theorem in a fancy disguise. If you can find the length and height of a triangle on the grid, you can find the distance.

What if a point has a negative number, like

(- 3, 4)

No problem! A negative x-number just means you walk left instead of right. A negative y-number means you climb down instead of up.

Is there a penalty if I guess on a hard coordinate geometry question?

Nope! The ISEE does not punish you for wrong answers. If you get stuck, pick your favorite letter and move on. Never leave a bubble blank!

What's the difference between Quantitative Reasoning and Math Achievement on the ISEE?

Quantitative Reasoning asks you to think outside the box and compare values (like the Column A vs. Column B questions). Math Achievement is more straightforward math, like finding an exact midpoint.

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