Choose a Point That Completes a Geometric Figure on a Grid - ISEE Lower Level Mathematics Achievement

SECTION 1

Why Do We Use Grids?

People have been drawing shapes on grids for thousands of years! Ancient builders in Egypt needed a way to plan their famous pyramids. They used ropes and stakes in the sand to create straight lines and perfect corners.

A grid (a pattern of squares made by crossing lines) helps us place points exactly where we want them. Think of it like graph paper in your notebook. The grid lines help you draw shapes that are neat and accurate.

3000 BC • Ancient Egypt

Builders used rope grids to plan pyramids with perfect square bases.

300 BC • Euclid's Geometry

A Greek mathematician named Euclid wrote rules about shapes like squares, rectangles, and triangles.

1600s • Coordinate Grids

René Descartes invented the numbered grid we use today. It lets us describe any point using two numbers!

Today • ISEE Test Questions

You use grids on math tests to find missing points that complete shapes. Let's learn how!

On the ISEE, you might see three corners of a rectangle on a grid. Your job is to find where the fourth corner goes. This lesson will teach you exactly how to do that. You've got this!

SECTION 2

Core Principles: What Makes a Shape a Shape?

Before we find missing points, let's review the shapes you'll see on the ISEE. Every shape has special rules. When you know the rules, finding that missing point becomes much easier!

Rectangles Have 4 Right Angles

A rectangle has four corners that are all perfect 90° angles (like the corner of a book). Opposite sides are the same length.

Squares Are Special Rectangles

A square is a rectangle where ALL four sides are the same length. Every square is a rectangle, but not every rectangle is a square!

Triangles Have 3 Corners

A triangle has exactly 3 sides and 3 corners (called vertices). Some triangles have a right angle, and some don't.

Parallelograms Lean Over

A parallelogram looks like a rectangle that got pushed to the side. Opposite sides are parallel (they never cross) and equal in length.

✦

KEY TAKEAWAY

Think of a grid like a treasure map. Each point has an exact address — how far right and how far up. If three friends are standing at three corners of a rectangle and you need to find where the fourth friend should stand, the grid tells you exactly where to go!

SECTION 3

See It on the Grid

Let's look at a grid with three points already placed. Can you tell where the missing fourth point of the rectangle should go? Study the diagram below!

Three corners of a rectangle are shown as solid dots at A (2, 6), B (8, 6), and C (8, 2). The missing corner D must be at (2, 2) to complete the rectangle. Notice the dashed green lines showing where D goes!

Look at the diagram above. Points A and B are both at height 6, so the top side is flat. Points B and C are both at position 8 going across, so the right side goes straight down. To finish the rectangle, point D needs to line up with both A (going down) and C (going left). That means D is at (2, 2)!

SECTION 4

The Math Behind Finding Missing Points

On a grid, every point has two numbers that tell you where it is. The first number tells you how far to go across (left to right). The second number tells you how far to go up (bottom to top). We write them like this: (across, up).

RECTANGLE RULE — OPPOSITE SIDES MATCH

If top side goes from (2, 6) to (8, 6), then bottom side goes from (2, 2) to (8, 2).

The "across" numbers for the left side match (both 2). The "across" numbers for the right side match (both 8). The "up" numbers for the top match (both 6). The "up" numbers for the bottom match (both 2).

MATCHING TRICK FOR RECTANGLES

Missing point = (across number from the point ABOVE or BELOW, up number from the point LEFT or RIGHT)

Look at the point directly above or below the missing corner — borrow its "across" number. Look at the point on the same level — borrow its "up" number. Combine them and you have your answer!

💡

ISEE Test Tip!

On the ISEE, always check your answer by imagining the finished shape. Does it look like a proper rectangle (or square, or triangle)? If one side looks crooked or a corner looks wrong, try a different answer choice!

SECTION 5

Different Shapes, Different Strategies

The ISEE can ask about different shapes. Let's look at how to find missing points for the most common ones. The diagram below shows three shape types you might see.

Three common shapes on the ISEE: a rectangle (left), a square (middle), and a right triangle (right). In each one, the green dashed circle marks the missing point you need to find.

Shape	How Many Corners?	Strategy to Find Missing Point
Rectangle	4	Match the "across" number from the point above/below. Match the "up" number from the point beside it.
Square	4	Same as rectangle, but also check that all sides are the same length.
Right Triangle	3	Find the point that makes one corner a perfect right angle (like the corner of a book).

Common shapes and how to find their missing points

SECTION 6

Worked Example: Finding the Missing Corner

Let's work through a full problem together, step by step. Take your time and follow along!

ℹ

Sample Problem

Three corners of a rectangle are at (1, 3), (1, 7), and (5, 7). Where is the fourth corner?

Finding the Fourth Corner

1
Step 1 — Plot what you know

Write down the three points: (1, 3), (1, 7), and (5, 7). Picture them on a grid. Two points share the "across" number 1. Two points share the "up" number 7.
2
Step 2 — Find matching "across" numbers

Points (1, 3) and (1, 7) both have 1 as the "across" number. That means they form a vertical side (going up and down) on the left.

Left side: from (1, 3) up to (1, 7)
3
Step 3 — Find matching "up" numbers

Points (1, 7) and (5, 7) both have 7 as the "up" number. They form a horizontal side (going left to right) at the top.

Top side: from (1, 7) across to (5, 7)
4
Step 4 — Find the missing point

The missing point must line up with (5, 7) going down — so its "across" number is 5. It must also line up with (1, 3) going right — so its "up" number is 3. Put them together!

The missing corner is at (5, 3). ✓
5
Step 5 — Check your answer

Does (1, 3), (1, 7), (5, 7), (5, 3) make a rectangle? Top side: 4 units wide. Bottom side: 4 units wide. ✓ Left side: 4 units tall. Right side: 4 units tall. ✓ It's actually a square! (A square is a special rectangle, so that's fine.)

All sides check out. Great job!

SECTION 7

ISEE Tips and Common Mistakes

Let's look at what helps and what trips people up on this type of problem. Knowing common mistakes helps you avoid them!

✅ Smart Strategies	❌ Common Mistakes
Look for numbers that appear twice — they tell you which points share a side.	Mixing up the "across" and "up" numbers. Always read across first, then up!
Draw lines between the three given points to see the shape forming.	Picking a point that makes a slanted side when the shape needs straight sides.
Check your answer: do opposite sides have equal length?	Forgetting to check that the final shape is the right type (rectangle vs. random quadrilateral).
Use process of elimination — test each answer choice on the grid.	Rushing and not plotting the points. Take a few seconds to visualize!

Tips vs. common mistakes for grid problems

🎯

TEST-TAKING STRATEGY

If you're stuck, try each answer choice! Imagine putting each point on the grid and drawing lines to the other three points. The correct answer will make the shape look right. On the ISEE, there's no penalty for guessing, so always pick an answer — never leave it blank!

SECTION 8

From Grids to Bigger Ideas

Finding points on a grid is the beginning of something really cool called coordinate geometry (using numbers to describe shapes). In later grades, you'll use this skill to do even more amazing things!

What You Learn Now	What Comes Later
Finding a missing corner of a rectangle	Finding missing corners of pentagons and hexagons
Counting grid squares to measure sides	Using formulas to calculate distance between any two points
Drawing rectangles and squares on grids	Graphing circles, curves, and other shapes on coordinate planes

How this skill grows as you advance in math

For now, mastering this skill will help you on the ISEE. Every point you find builds your confidence with shapes and numbers. You're building a strong math foundation!

SECTION 9

Practice Problems

Now it's your turn! Try these five problems. They start easy and get a little harder. Remember: look for matching numbers and always check your answer!

Question 1: CONCEPTUAL

Three corners of a rectangle are at (1, 1), (1, 4), and (5, 4). Where is the fourth corner?

A. (3, 3)
B. (4, 1)
C. (5, 1)
D. (5, 5)

Show answer (C)

The correct answer is (5, 1). Points (1, 4) and (5, 4) share the "up" number 4, forming the top side. Points (1, 1) and (1, 4) share the "across" number 1, forming the left side. The missing point needs the "across" number from (5, 4) which is 5, and the "up" number from (1, 1) which is 1. That gives us (5, 1). Choice A (3, 3) is in the middle, not at a corner. Choice B (4, 1) has the wrong "across" number. Choice D (5, 5) has the wrong "up" number.

Question 2: BASIC CALCULATION

Three corners of a square are at (2, 2), (2, 6), and (6, 6). Where is the fourth corner?

A. (4, 4)
B. (6, 2)
C. (6, 4)
D. (8, 2)

Show answer (B)

The correct answer is (6, 2). The left side goes from (2, 2) to (2, 6) — that's 4 units tall. The top side goes from (2, 6) to (6, 6) — that's 4 units wide. Since it's a square, all sides are 4 units. The missing point needs the "across" number 6 (matching the top-right corner) and the "up" number 2 (matching the bottom-left corner). That gives (6, 2). Choice A (4, 4) is the center of the square, not a corner. Choice C (6, 4) would make the right side only 2 units, not 4. Choice D (8, 2) would make the bottom side 6 units, which is too long.

Question 3: INTERMEDIATE

Three corners of a rectangle are at (3, 5), (7, 5), and (7, 9). Which point completes the rectangle?

A. (3, 7)
B. (3, 9)
C. (5, 9)
D. (7, 3)

Show answer (B)

The correct answer is (3, 9). Points (3, 5) and (7, 5) share the "up" number 5, forming the bottom side. Points (7, 5) and (7, 9) share the "across" number 7, forming the right side. The fourth point needs the "across" number from (3, 5) which is 3, and the "up" number from (7, 9) which is 9. That gives us (3, 9). Choice A (3, 7) has the right "across" number but the wrong "up" number. Choice C (5, 9) has the wrong "across" number. Choice D (7, 3) would be below the rectangle, not completing it.

Question 4: APPLIED

Maya is drawing a rectangle on grid paper. She has placed three corners at (2, 1), (2, 8), and (6, 1). She wants to place the last corner so the rectangle is complete. What should the last corner be?

A. (4, 4)
B. (6, 4)
C. (6, 8)
D. (8, 8)

Show answer (C)

The correct answer is (6, 8). Points (2, 1) and (2, 8) form the left side (both have "across" number 2). Points (2, 1) and (6, 1) form the bottom side (both have "up" number 1). The missing top-right corner needs the "across" number 6 (matching the bottom-right point) and the "up" number 8 (matching the top-left point). That makes (6, 8). Let's check: the top side from (2, 8) to (6, 8) is 4 units, and the bottom side from (2, 1) to (6, 1) is 4 units. ✓ Choice A (4, 4) is in the middle. Choice B (6, 4) is too low. Choice D (8, 8) is too far right.

Question 5: CRITICAL THINKING

Two corners of a right triangle are at (1, 2) and (1, 7). The right angle is at (1, 2). Which of these points could be the third corner to complete the right triangle?

A. (1, 5)
B. (3, 5)
C. (5, 2)
D. (7, 7)

Show answer (C)

The correct answer is (5, 2). A right angle means two sides meet at a perfect 90° corner (like the letter L). The right angle is at (1, 2). One side goes straight up from (1, 2) to (1, 7). For the right angle to work, the other side must go straight across from (1, 2). Only (5, 2) shares the "up" number 2 with (1, 2), creating a horizontal side. That makes the two sides at (1, 2) form an L shape — a perfect right angle! Choice A (1, 5) is on the same vertical line, so it wouldn't form a triangle at all. Choice B (3, 5) would create a slanted side at the right-angle corner, not a true right angle. Choice D (7, 7) also creates a slanted side, not a right angle.

SUMMARY

Let's Review!

Summary

To find a missing point that completes a shape on a grid, start by looking at the points you already have. Find matching numbers — two points that share an "across" number form a vertical side, and two points that share an "up" number form a horizontal side. For rectangles and squares, borrow the "across" number from the point above or below and the "up" number from the point beside it.

Always check your answer by making sure opposite sides are equal and the shape looks right. If you're stuck, use process of elimination — try each answer choice on the grid and see which one makes the correct shape. Remember, on the ISEE there's no penalty for guessing, so always pick an answer. You've got this!