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⬅ Back to Plane Geometry
📏 Key Concepts: Similar Figures & Other Shapes

Similar Triangles: Two triangles are similar (AA, SAS, SSS similarity) if their corresponding angles are equal and sides are proportional.

Ellipse:

Problem 1: Similar Triangles

Triangle ABC is similar to triangle DEF. If AB = 6 cm, BC = 8 cm, AC = 10 cm, and DE = 9 cm, find:
a. The lengths of EF and DF
b. The area of triangle DEF, given that the area of triangle ABC is 24 cm²

Plane Geometry – Similar Figures – Problem 1 – Diagram Plane Geometry – Similar Figures – Problem 1 – Diagram Plane Geometry – Similar Figures – Problem 1 – Diagram
Solution Solution Solution Solution

Step 1 — Find the scale factor:

$$k = \frac{DE}{AB} = \frac{9}{6} = \frac{3}{2}$$

a. Missing sides:

$$EF = k \cdot BC = \frac{3}{2}(8) = \boxed{12 \text{ cm}}, \qquad DF = k \cdot AC = \frac{3}{2}(10) = \boxed{15 \text{ cm}}$$

b. Area of DEF (area scales as $k^2$):

$$A_{\triangle DEF} = k^2 \cdot A_{\triangle ABC} = \left(\frac{3}{2}\right)^2 \times 24 = \frac{9}{4} \times 24 = \boxed{54 \text{ cm}^2}$$

Problem 2: Ellipse — Area and Perimeter

An ellipse has a semi-major axis of $a = 10$ cm and a semi-minor axis of $b = 6$ cm. Find:
a. The area of the ellipse
b. The approximate perimeter using Ramanujan's formula: $P \approx \pi\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]$

Plane Geometry – Similar Figures – Problem 2 – Diagram Plane Geometry – Similar Figures – Problem 2 – Diagram Plane Geometry – Similar Figures – Problem 2 – Diagram
Solution Solution Solution Solution

a. Area:

$$A = \pi ab = \pi(10)(6) = 60\pi \approx \boxed{188.5 \text{ cm}^2}$$

b. Approximate perimeter using Ramanujan's formula:

$$P \approx \pi\!\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]$$
$$= \pi\!\left[3(16) - \sqrt{(30+6)(10+18)}\right] = \pi\!\left[48 - \sqrt{36 \times 28}\right] = \pi\!\left[48 - \sqrt{1008}\right]$$
$$= \pi\!\left[48 - 31.75\right] = \pi(16.25) \approx \boxed{51.1 \text{ cm}}$$