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! ⭐
Practice Questions
4 practice questions for ISEE Lower Level
(2, 1), (2, 6), and (8, 1) on a coordinate grid. What are the coordinates of the fourth vertex of the rectangle?(1, 6) (6, 8) (8, 6) (8, 8) Show Solution
- A rectangle has opposite sides that are parallel and equal in length, meaning its vertices will share x-coordinates and y-coordinates. The given points have x-coordinates of 2 and 8, and y-coordinates of 1 and 6. The fourth vertex must use the remaining x-coordinate (8) and the remaining y-coordinate (6) to close the shape. Therefore, the coordinates of the fourth vertex are
(8, 6).
A(3, 2), B(5, 6), C(7, 6), and D(9, 2) are connected in order to form a polygon. Which term best describes the polygon formed?Show Solution
- One way to find the answer is to analyze the coordinates of the sides. Side connects
(5, 6)and(7, 6), which is a horizontal line segment. Side connects(3, 2)and(9, 2), which is also a horizontal line segment. Because both of these sides are horizontal, they are parallel to each other. The other two sides, and , are not parallel. A quadrilateral with exactly one pair of parallel sides is a trapezoid.
(4, 7). The piece is moved 3 units to the left and 2 units down. What are the new coordinates of the game piece?(1, 5) (1, 9) (7, 5) (7, 9) Show Solution
- Moving 3 units to the left means subtracting 3 from the x-coordinate: . Moving 2 units down means subtracting 2 from the y-coordinate: . The new coordinates of the game piece are
(1, 5).
P(3, 8) to point Q(3, 2). What is the length of the line segment ?Show Solution
- Both points have the same x-coordinate of 3, which means the line segment is a vertical line. To find the length of a vertical line segment, subtract the smaller y-coordinate from the larger y-coordinate: . The length of the line segment is 6 units.
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 ().
- 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' (). 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 ?
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.