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Lesson 3 of 12

Area · Lesson 3 of 12

Using Triangle Area to Solve Problems

“Use equal-area reasoning, alternative base-height pairs, rectangle diagonals, medians, and midpoint constructions to solve geometric problems.”

Learning Objectives

• Use two different base-height pairs for one triangle to find an unknown length. • Recognise when triangles have equal areas even if they are not congruent. • Explain why the diagonals of a rectangle create four equal-area triangles. • Show why a median divides a triangle into two equal-area parts. • Solve midpoint and area-fraction problems by reasoning with common heights and proportional bases.

Knowing a formula is only the beginning. Triangle area becomes a powerful reasoning tool when the area can be written in more than one way. Instead of measuring every length directly, we can compare bases and heights, exploit midpoints, and use equal-area relationships.

This lesson focuses on structures hidden inside diagrams. The central question is often not “Which formula should I use?” but “Which two triangles share a useful height or base?”

One Triangle, Two Base-Height Pairs

Suppose a triangle has one known base-height pair, so its area can be calculated. If a different side length is also known, the same numerical area can be expressed using that side as a new base. Equating the two area expressions reveals the corresponding unknown height.

Same triangle, same areaLaTeX
The halves cancel, giving b₁h₁ = b₂h₂.
Finding a second altitude

Problem
A triangle has one base 5 units with height 3 units. Another side has length 4 units. Find the height to that side.

  1. 1.Using the first pair, area = ½ × 5 × 3 = 15/2 square units.
  2. 2.Using the 4-unit side as base, area = ½ × 4 × h = 2h.
  3. 3.Set 2h = 15/2.
  4. 4.h = 15/4 = 3.75 units.

Equal Area Does Not Mean Congruent

Congruent triangles certainly have equal areas, but the reverse is not true. A tall narrow triangle and a low wide triangle can cover the same amount of surface. Equal area follows whenever the product base × corresponding height is equal.

Definition
Equal-area figures

Figures that cover the same amount of surface. They may have different side lengths, angles, shapes, or perimeters.

Different shapes, equal area

Problem
Triangle A has base 10 cm and height 6 cm. Triangle B has base 15 cm and height 4 cm. Compare their areas.

  1. 1.Triangle A: ½ × 10 × 6 = 30 cm².
  2. 2.Triangle B: ½ × 15 × 4 = 30 cm².
  3. 3.They have equal areas even though their base and height measurements differ.

Four Triangles Inside a Rectangle

Draw both diagonals of a rectangle and let them intersect at O. The diagonals bisect each other, so OB = OD and OA = OC. Consider two adjacent triangles that use OB and OD as bases. Their third vertices lie on the same side of the rectangle, so they have the same altitude to the diagonal line. Equal bases plus equal heights give equal areas.

Repeating the argument around the intersection shows that all four triangles formed by the two diagonals have equal area. They need not all be congruent in a non-square rectangle; equal area comes from the base-height relationship.

Each diagonal triangle in a rectangleLaTeX
Area of one diagonal region

Problem
A rectangle has dimensions 12 cm by 8 cm. Both diagonals are drawn. Find the area of each of the four triangles around their intersection.

  1. 1.Rectangle area = 12 × 8 = 96 cm².
  2. 2.The diagonals divide the rectangle into four equal-area triangles.
  3. 3.Each area = 96 ÷ 4 = 24 cm².

A Median Divides Area in Half

Definition
Median of a triangle

A line segment joining a vertex of a triangle to the midpoint of the opposite side.

Let M be the midpoint of side BC in triangle ABC. Then BM = MC. Triangles ABM and ACM use equal bases BM and MC along the same line, and both share the same perpendicular height from A to line BC. Their areas are therefore equal.

Median area relationLaTeX
Median with a known total area

Problem
Triangle PQR has area 54 cm². M is the midpoint of QR. Find the areas of triangles PQM and PRM.

  1. 1.PM is a median because M is the midpoint of QR.
  2. 2.A median creates two triangles with equal bases and the same height.
  3. 3.Each area is 54 ÷ 2 = 27 cm².

Two Midpoints and Area Fractions

Now let M and N be the midpoints of XY and XZ in triangle XYZ. Segment XN first divides the triangle into two equal-area triangles because N is the midpoint of XZ. Inside triangle XYN, segment NM can be analysed using M as the midpoint of XY. Repeated halving of bases produces a smaller triangle whose area is one quarter of the original.

Midpoint triangle at a vertexLaTeX
Both sides from X are halved, and the resulting similar triangle has length scale 1/2 and area scale 1/4.
Area fraction from two midpoints

Problem
Triangle XYZ has area 80 cm². M and N are the midpoints of XY and XZ. Find the area of triangle XMN.

  1. 1.Joining midpoint points halves both sides from X.
  2. 2.The small triangle has one quarter of the original area.
  3. 3.Area = 80 × 1/4 = 20 cm².

Reasoning Before Calculating

When a diagram contains midpoints, diagonals, or parallel lines, mark equal segments and shared heights before reaching for arithmetic. A short structural argument often removes the need for unknown side lengths entirely.

Equal Area Is Not the Same as Equal Shape

Do not claim two regions are congruent merely because their areas are equal. Congruence requires matching shape and size, while equal area only compares surface measure.

Quiz

Quick check

Why can the same triangle's area be written using different bases?

Quick check

Two triangles have equal bases and equal corresponding heights. Their areas are

Quick check

The diagonals of a rectangle divide it into how many equal-area triangles?

Quick check

A median divides a triangle into two regions whose areas are

Quick check

If two sides from one vertex are both halved by taking midpoints, the small corner triangle has what fraction of the original area?

Practice Problems

Practice Problems
  1. A triangle has base 8 cm and height 9 cm. A second side is 12 cm. Find the altitude to the 12 cm side.
  2. Two triangles have dimensions (base, height) = (18 cm, 5 cm) and (10 cm, 9 cm). Compare their areas.
  3. A 15 cm by 10 cm rectangle has both diagonals drawn. Find the area of each of the four triangles.
  4. A triangle has area 96 cm². A median is drawn. Find the area of each part and explain why.
  5. In triangle XYZ, M and N are midpoints of XY and XZ. If triangle XMN has area 17 cm², find the area of triangle XYZ.
  6. Create two non-congruent triangles of area 24 cm² using different base-height pairs.
  7. Explain why equal area alone cannot prove congruence.

Key Takeaways

Key Takeaways

• A triangle's area can be expressed with any side and its corresponding perpendicular height. • Equating two area expressions can reveal an unknown altitude. • Equal-area triangles need not be congruent. • The diagonals of a rectangle form four equal-area triangles. • A median halves a triangle's area because it creates equal bases with a common height. • Midpoints often create simple area fractions without requiring all side lengths.

Previous · Lesson 2

Area of a Triangle

Next · Lesson 4

Triangles Between Parallel Lines