Lesson 3 — Triangles
Big idea: A triangle’s area is half the product of a base and its corresponding altitude, and a right triangle is just the special case where the legs serve as base and altitude.
Words to know
| Word | Meaning |
|---|---|
| Base of a triangle | Any one side chosen as the base. |
| Altitude of a triangle | The perpendicular distance from a vertex to the line of the opposite side; also the perpendicular segment itself. |
| Corresponding altitude | The altitude drawn to the chosen base. |
| Legs (right triangle) | The two sides that form the right angle. |
| Hypotenuse | The side opposite the right angle in a right triangle. |
Concepts to know
- An altitude may lie on a side, inside the triangle, or outside the triangle.
- Theorem 38: The area of a right triangle is half the product of its legs.
- Theorem 39: The area of a triangle is half the product of any base and corresponding altitude.
- Corollary to Theorem 39: Triangles with equal bases and equal altitudes have equal areas.
- The corollary works even when the triangles are not congruent, such as an acute and an obtuse triangle with the same base and height.
- An altitude creates right triangles that connect the triangle’s area to a related rectangle, which is how the general formula is proved.
Formulas / rules
- General triangle: A = ½bh
- Right triangle: A = ½(leg₁)(leg₂)
- The base and altitude must correspond: the altitude is perpendicular to the chosen base.
- Same area does not mean same shape or same perimeter.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find a triangle’s area | Choose a base, find the perpendicular altitude to it, then use A = ½bh. |
| Find a missing height | Substitute the area and base into A = ½bh and solve for h. |
| Find a missing base | Solve A = ½bh for b. |
| Find the area of a right triangle | Multiply the two legs and divide by 2. |
| Compare two triangles’ areas | Compare ½bh for each, or use the equal bases/altitudes corollary. |
| Work with an obtuse triangle | Extend the base if needed; the altitude may fall outside. |
Watch out
- The altitude must be perpendicular to the base you use.
- Do not use a slanted side as the height.
- For a right triangle, the hypotenuse is not a leg; use the two legs in Theorem 38.
- Equal areas do not imply congruent triangles.
- Do not forget the ½.