GANITA MANJARI · GRADE 9 · PART I · CHAPTER 6

TRIANGLE — Discovering Triangle Area

Think first. Use AI purposefully. Verify always.

DISCOVER

Start with a question.

How can two congruent copies of a triangle make an area you already know?

Understand

Two copies form a parallelogram, so triangle area is half base times perpendicular height. A median splits a triangle into equal areas because the two bases are equal and share an altitude.

Key Ideas

A = \frac{bh}{2}

Equal areas do not imply congruent triangles. A median is not generally perpendicular to the base.

Visualise

Can a second congruent triangle complete a parallelogram?

The copy is a half-turn about the midpoint of the diagonal.

Apex horizontal position: 120

The equal base marks and common altitude justify equal areas; the triangles need not be congruent.

Method

Choose a base and its perpendicular height (possibly outside the triangle). A median joins a vertex to the midpoint of the opposite side.

Worked Example

Given

BC = 10 cm; altitude from A is 6 cm; D is the midpoint of BC.

To Find

Find areas ABC, ABD and ACD.

Formula / Property / Theorem / Reasoning

Equal bases BD and DC share the altitude from A.

Solution

Area ABC = $\frac{(10 \times 6)}{2}$ = 30 ${cm}^{2}$ .
BD = DC = 5 cm. Each smaller area is $\frac{(5 \times 6)}{2}$ .

Result

Area ABD = area ACD = 15 ${cm}^{2}$ . Original AIMS example.

AIMS Method

A — Attempt independently. Make a prediction or start a sketch.
I — Investigate the idea. Identify the boundary, region and given conditions.
M — Make connections. Relate the figure to a known property or rearrangement.
S — Scrutinise your reasoning. Check conditions, units and whether the result is exact.

Think first. Use AI purposefully. Verify always.

Focused Practice

A triangle has base 12 m and height 5 m. A median meets that base. Find each smaller area.

Your reasoning

Check & Feedback

Does a median always create two congruent triangles?

Explain your conclusion

Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 6 · 6.8 · Figures 6.20–6.22 · printed pages 132–134.