Right Triangles in Coordinates

Overview

Students apply Pythagoras’ theorem to find lengths in right-angled triangles and connect this to coordinate geometry by calculating distances on a grid. The lesson builds from recognising right angles and labelling triangle sides to using a coordinate method to solve problems.

Learning intentions

• Students will identify the right angle and the perpendicular sides in a triangle on a coordinate grid.
• Students will use Pythagoras’ theorem to calculate unknown side lengths.
• Students will find distances between points by forming a right-angled triangle from coordinate differences.
• Students will justify their calculations using correct substitution into Pythagoras’ theorem.

Success criteria

• I can state Pythagoras’ theorem and correctly substitute values for (a), (b), and (c).
• I can determine which two sides are perpendicular when given coordinates.
• I can calculate the distance between two points on a grid and give the exact or rounded value as required.
• I can explain how the coordinate differences form a right-angled triangle.

Curriculum links

• MA4-PYT-C-01 — Students apply Pythagoras’ theorem to solve problems in various contexts (right-angled triangles).
• MALS-GEO-01 — Students explore 2-dimensional shapes and 3-dimensional objects (features of 2D space; coordinates).
• MA4-GEO-C-01 — Students identify and apply properties of triangles to solve problems (classifying right-angled triangles by their side/angle features).

Lesson structure ({total minutes})

1. 0–6 min · Hook (notice and name). Teacher shows a coordinate grid with two points (e.g., A(1,2) and B(7,2)) and a third point C(7,6), forming a right-angled triangle, and asks students to quickly discuss: “Which segment is the hypotenuse?” Students label the right angle and predict which side is longest using the diagram.

2. 6–16 min · Mini-teach (Pythagoras in coordinates). Teacher explicitly connects coordinate differences to triangle side lengths: horizontal change = (|\Delta x|), vertical change = (|\Delta y|), and the distance is the hypotenuse. Students copy a worked example: distance between A(1,2) and C(7,6). They compute (|\Delta x|=6), (|\Delta y|=4), then (d=\sqrt{6^2+4^2}=\sqrt{52}\approx 7.2).

3. 16–30 min · Guided practice (three short tasks). Teacher circulates while students solve in pairs on grid paper or a worksheet, with prompts to check perpendicular sides and correct substitution order. Students complete:

• Task 1: Find the missing length in a right triangle with vertices A(0,0), B(6,0), C(6,8).
• Task 2: Find the distance between two points that share neither x nor y, e.g., D(2,1) and E(9,5).
• Task 3: Determine whether a triangle with points F(0,0), G(5,0), H(2,4) is right-angled by comparing sides using Pythagoras (decide using exact values if possible).

4. 30–42 min · Whole-class check (error-hunting). Teacher presents two incorrect solutions (e.g., students accidentally using (|\Delta x|) and (|\Delta y|) swapped, or treating the hypotenuse as a perpendicular side) and asks, “What mistake happened and how do we fix it?” Students identify the error and correct the method, using the success criteria language.

5. 42–55 min · Independent application (distance on a grid). Teacher gives a clear problem set on coordinates requiring distance calculations and one multi-step justification. Students work individually, then confirm one answer with a partner. Students solve:

• Calculate the distance between P(3,7) and Q(11,3).
• Calculate the length of the diagonal of a rectangle with corners R(1,1), S(1,9), T(10,9), U(10,1) using coordinate differences.
• Justify in one paragraph: “Explain how coordinate geometry lets us use Pythagoras’ theorem to find a distance.”

6. 55–60 min · Exit ticket (quick formative assessment). Teacher collects an exit ticket with one short item: Given A(4,2) and B(9,10), find the distance and show which side lengths correspond to (|\Delta x|) and (|\Delta y|). Students submit their working and final answer (exact surd or rounded to the nearest tenth as instructed).

Resources

• Coordinate grid worksheets (A4) with clear scales
• Pencils, erasers, ruler for straight lines
• Worked example card (horizontal/vertical difference → Pythagoras substitution)
• Individual mini-whiteboards or scrap paper for checking hypotenuse identification
• Teacher pre-prepared “error-hunting” solutions on a slide or board
• Timer for timed practice prompts
• Optional: calculators for decimal approximations (not for the substitution step)

Assessment

• Formative check during guided practice: teacher listens for correct identification of perpendicular sides and correct substitution into (a^2+b^2=c^2).
• Formative check in whole-class error-hunt: students explain the mistake using correct triangle language (perpendicular sides, hypotenuse).
• Exit ticket: accuracy of distance calculation and clarity about how (|\Delta x|) and (|\Delta y|) form the right-angled triangle.

Differentiation

• Support: Provide sentence starters for justification (“I found (|\Delta x|=\dots) and (|\Delta y|=\dots), so the perpendicular sides are … therefore …”).
• Support: Offer a step-by-step scaffold sheet showing the process: compute differences → square → add → square root → interpret as distance.
• Extension: Students solve an “inverse” problem: given a distance, find possible points on the grid with matching (|\Delta x|) and (|\Delta y|), then explain how you know.
• EAL/SEN: Use colour-coding on the grid for horizontal and vertical segments; allow use of a calculator only after the setup is correct.