The Life Starter
Sign up
Grade 10 · lesson 4 of 4

Use Perpendicular Coordinate-Axis Legs

A triangle with legs along the coordinate axes has an explicit right angle at their meeting point.

About 20 minutes, plus any extended practice

Download the grade pack

Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: A triangle has vertices (0,0), (4,0), and (0,3). Find its area. Candidate answers: (A) 12 square units (B) 5 square units (C) 6 square units (D) 7 square units.
  • Independent check inputs: A triangle has vertices (0,0), (6,0), and (0,4). Find its area. Candidate answers: (A) 11 square units (B) 12 square units (C) 13 square units (D) 24 square units.

Learn the idea

Geometric properties can be expressed through coordinates as well as diagrams. Parallelogram angle and diagonal theorems, midpoint coordinates, perpendicular slopes and perpendicular base-height relationships provide exact arguments about figures. A triangle with legs along the coordinate axes has an explicit right angle at their meeting point. Its perpendicular base and height determine area as half their product.

Here is the method on the supplied teaching case. The axes meet at a right angle, so the base 4 and perpendicular height 3 give A=(1/2)bh=6. The triangle is half its bounding rectangle.

A slanted side is not automatically the height. Identify a perpendicular distance to the chosen base; coordinate differences supply the needed lengths in this stated axis-aligned triangle.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: A triangle has vertices (0,0), (4,0), and (0,3). Find its area. Candidate answers: (A) 12 square units (B) 5 square units (C) 6 square units (D) 7 square units. Solution: The axes meet at a right angle, so the base 4 and perpendicular height 3 give A=(1/2)bh=6. The triangle is half its bounding rectangle. The final answer is 6 square units. A slanted side is not automatically the height. Identify a perpendicular distance to the chosen base; coordinate differences supply the needed lengths in this stated axis-aligned triangle.

Your turn

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Helpful tips
  • A triangle with legs along the coordinate axes has an explicit right angle at their meeting point.
  • A slanted side is not automatically the height.
A triangle has vertices (0,0), (6,0), and (0,4). Find its area.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

12 square units

The axes meet at a right angle, so the base 6 and perpendicular height 4 give A=(1/2)bh=12. The triangle is half its bounding rectangle.

Take it further

Changed-input task: A triangle has vertices (0,0), (18,0), and (0,10). Find its area. Candidate answers: (A) 180 square units (B) 89 square units (C) 90 square units (D) 91 square units. Show the mathematical reason, not only the selected answer. Expected result: 90 square units. The axes meet at a right angle, so the base 18 and perpendicular height 10 give A=(1/2)bh=90. The triangle is half its bounding rectangle.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. A triangle has vertices (0,0), (8,0), and (0,5). Find its area.
Need a hint?
  • A triangle with legs along the coordinate axes has an explicit right angle at their meeting point.
  • A slanted side is not automatically the height.
2. A triangle has vertices (0,0), (10,0), and (0,6). Find its area.
Need a hint?
  • A triangle with legs along the coordinate axes has an explicit right angle at their meeting point.
  • A slanted side is not automatically the height.
3. A triangle has vertices (0,0), (12,0), and (0,7). Find its area.
Need a hint?
  • A triangle with legs along the coordinate axes has an explicit right angle at their meeting point.
  • A slanted side is not automatically the height.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.