Depth · one concept from first principles · step 2 of 4

Use shared base to find composite area

In the figure at right, triangle ABC and triangle DEF have the same shape and size, and triangle AGD and triangle GEC also have the same shape and size. What is the area of the shaded region, in $\text{cm}^2$ ?

Figure description: Two right triangles overlap. Along the bottom, the points B, E, C, F lie in that order on one straight line. The larger left right triangle is ABC, and the angle at vertex A measures $45^\circ$ . The right-hand right triangle is DEF; its top side AD is $4\,\text{cm}$ long and its right side DF is $8\,\text{cm}$ long. The point where the two triangles cross is labeled G, and right angles are formed at C and F where the slanted sides meet the base.

GeometryMeasurement & data

Your answer

How to solve

Strategy Identify Subproblems — The shaded shape is a compound figure built from two overlapping triangles, so I break it into pieces I can measure: each whole triangle, and the overlap they share. Drawing and labeling the picture with coordinates makes the equal legs and the overlap visible. Because the overlap is exactly triangle GEC, the shaded area equals one triangle plus the rest of the other triangle. I prefer plain subproblem-and-add-up reasoning over algebra because the lengths are all directly readable from the figure.

1STEP 1

Find each leg length

A right angle and a 45 degree angle force the third to 45, so the legs match and each is 8 cm.

angle B = 180° - 90° - 45° = 45°

2STEP 2

Area of one whole triangle

Each triangle has perpendicular legs of 8 cm, and a triangle's area is half of base times height.

1/2 × 8 × 8 = 32 cm²

3STEP 3

Area of the overlap

The overlap is triangle GEC, whose perpendicular sides are 4 cm each, so its area is 8 square cm.

1/2 × 4 × 4 = 8 cm²

4STEP 4

Add the two triangles and subtract the shared overlap

Adding both triangles counts the overlap twice, so take it off once: 32 + 32 - 8.

32 + 32 - 8 = 56 cm²

Answer

56 cm²

Each triangle alone is 32 square centimeters, so two of them with a small shared corner should be a bit less than 64. The answer 56 is just below 64, which fits, and the units are square centimeters as required for area.

Takeaway

When two shapes overlap, add their areas and take the shared middle away once so you do not count it twice.

Find each leg length
Area of one whole triangle
Area of the overlap
Add the two triangles and subtract the shared overlap

Where next?

Level upsuggested

Solve known areas first, step by step Count how many unit shapes make the whole

Another one like thissuggested

12 practice problems

Step back and solidifysuggested

Identify tiling pieces and total area Count sticks needed via multiplication

A parent dashboard for the family lives at sensimlab.com.

▶ Practice — 12 problems