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Any Side Can Be the Base

Part of the Triangles section of Coddy's Geometry journey. Lesson 21 of 63.

Every triangle has three bases and three heights: pick any side as the base, and the height is the drop onto it from the opposite corner. A triangle has only one area, so every pair gives the same answer.

$Triangle ABC with height CD onto AB and height AE onto BC$

Here CD is the height onto AB, and AE is the height onto BC. With AB = 12 and CD = 10, the area is 1/2 × 12 × 10 = 60. The other pair must give 60 too: 1/2 × 15 × h = 60, so the height AE is 8.

That is a two-step chase on the stage: first the area from one base and its height, then the other height from the area. Pick Area of a triangle for both.

Challenge

△ABC has two heights drawn: CD onto AB and AE onto BC. Find the height AE, then press Check.

There is nothing to type: tap a side, or inside the triangle for its area, and pick the rule that gives it. The value then appears on the figure.

Try it yourself

$Find the marked part of the figure$

Find the marked part of the figure

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Triangles

1Sides and Names

Three Sides, Three Corners Sorting by Sides Naming by Angles Ticks into Equations The Longest Side Recap: Sorting Triangles

4Area of a Triangle

Half the Rectangle Height Is Not a Side Right Triangle Area Finding the HeightAny Side Can Be the BaseRecap: Area

7Is It Right?

The Converse Testing a Triangle Acute or Obtuse?Sorting by the Test Recap: Right or Not

10Scale in the World

Shadows Across the River Nested Triangles A Side with an Expression Recap: Scale in the World

2The Triangle Inequality

Can Three Sticks Meet?Exactly Reaching The Range of the Third Side The Formula Recap: The Third Side

5The Pythagorean Theorem

Squares on the Sides Finding the Hypotenuse Pythagorean Triples When It Is Not Whole Recap: The Hypotenuse

8Special Right Triangles

The 45-45-90 Triangle Legs from the Hypotenuse The 30-60-90 Triangle All Three Sides Which Triangle Is It?Recap: Special Triangles

11Two Tools at Once

Height of an Isosceles Area from the Height The Equilateral Triangle Right Triangle Perimeter Recap: Two Tools