Geometry Basics

Perimeter: $P = 2l + 2w = 2(12) + 2(5) = 24 + 10 = 34$

Area: $A = lw = 12 \times 5 = 60$

Answer: Perimeter = 34, Area = 60

When parallel lines are cut by a transversal, we get two types of angles:

The angle of 115° and its vertical angle are both 115°. The supplementary angles are $180° - 115° = 65°$.

All eight angles are either 115° or 65°:

• Four angles of 115° (the angle, its vertical angle, and corresponding angles)
• Four angles of 65° (supplementary to the 115° angles)

Answer: Four angles are 115° and four are 65°.

Key relationships used: vertical angles, corresponding angles, supplementary angles.

3D diagonal formula: $d = \sqrt{l^2 + w^2 + h^2}$

$d = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$

Answer: The space diagonal is 13.

Alternatively: First find the diagonal of the base: $\sqrt{3^2 + 4^2} = 5$ (3-4-5 triple), then use that with the height: $\sqrt{5^2 + 12^2} = \sqrt{169} = 13$ (5-12-13 triple).

Step 1: Find the radius using the volume formula: $V = \pi r^2 h$ $100\pi = \pi r^2 (4)$ $r^2 = 25$ $r = 5 \text{ cm}$

Step 2: Calculate total surface area (two circles + lateral surface): $SA = 2\pi r^2 + 2\pi rh$ $= 2\pi(25) + 2\pi(5)(4)$ $= 50\pi + 40\pi$ $= 90\pi \approx 282.74 \text{ cm}^2$

Answer: $90\pi$ cm² (approximately 282.74 cm²)

Key property of similar figures: If corresponding sides have ratio $k$, then areas have ratio $k^2$.

Side ratio: $\frac{3}{5}$

Area ratio: $\left(\frac{3}{5}\right)^2 = \frac{9}{25}$

$\frac{27}{A_{\text{large}}} = \frac{9}{25}$ $A_{\text{large}} = \frac{27 \times 25}{9} = 75 \text{ cm}^2$

Answer: 75 cm²

Remember: Side ratio = $k$, Area ratio = $k^2$, Volume ratio = $k^3$.