1. The two triangles are similar by the AA (Angle-Angle) similarity criterion because they share a 30° angle and both have a right angle (90°). The third angle must also be equal (60°), confirming similarity.
2. The scale factor from the smaller triangle to the larger triangle is 2, as the corresponding sides are in the ratio 4:8 or 5:10.
3. Using the Pythagorean theorem for the smaller triangle: $a^2 + b^2 = c^2$, where $a = 4$ and $c = 5$. Solving for $b$: $b = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$.
4. For the larger triangle, since the scale factor is 2, the missing side is $3 \times 2 = 6$.
5. The length of the platform is 17.5 feet, calculated using the Pythagorean theorem with legs 10.5 ft and 14 ft: $\sqrt{10.5^2 + 14^2} = \sqrt{110.25 + 196} = \sqrt{306.25} = 17.5$.
6. In the coordinate plane, the slope of line segment AB is $\frac{6 - 2}{4 - 1} = \frac{4}{3}$, and the slope of line segment CD is $\frac{8 - 4}{6 - 3} = \frac{4}{3}$, so the segments are parallel.
7. The distance between points A(1,2) and B(4,6) is $\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
8. The midpoint of segment AB is $\left(\frac{1+4}{2}, \frac{2+6}{2}\right) = (2.5, 4)$.
9. The area of the triangle with vertices at (0,0), (4,0), and (0,3) is $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 3 = 6$ square units.
10. The perimeter of the rectangle with vertices at (0,0), (5,0), (5,3), and (0,3) is $2 \times (5 + 3) = 16$ units.
Parent Tip: Review the logic above to help your child master the concept of angle angle similarity worksheet.
