NY Regents Geometry: a complete guide to similarity, trigonometry, and circles on the exam

What this strand demands

The similarity-trigonometry-circles strand carries much of the computation on NY Regents Geometry. It rewards fluent use of proportions (similarity), the trig ratios (right triangles), the circle angle and segment relationships, coordinate methods (distance, slope, partition), and the volume and density formulas. This guide ties together the dot-point pages, each with its own practice: dilations and similarity, right triangle trigonometry, circles, angles, and segments, coordinate geometry and partitioning, and volume and solids.

Dilations and similarity

A dilation with center and scale factor $k$ multiplies lengths by $k$ and keeps angles fixed, producing a similar figure. Two figures are similar if a sequence of rigid motions and a dilation maps one onto the other. Triangles are usually proved similar by AA; corresponding sides are then proportional, so a proportion finds a missing length. Lengths scale by $k$, areas by $k^2$.

Right triangle trigonometry

SOHCAHTOA defines the ratios: sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent. Pick the ratio that links the side you want to a side you know; use an inverse trig function to find an angle. The acute angles are complementary, so $\sin\theta = \cos(90^\circ - \theta)$. Angle of elevation looks up, angle of depression looks down.

Circles

A central angle equals its arc; an inscribed angle is half its arc (so a semicircle inscribes a right angle). A tangent is perpendicular to the radius at the point of contact. Arc length and sector area are the fraction $\frac{\theta}{360}$ of the circumference and area. The equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, with center $(h, k)$ (flip the signs) and radius $\sqrt{r^2}$.

Coordinate geometry and partitioning

Distance tests congruent segments, midpoint tests bisection, slope tests parallel (equal) and perpendicular (negative reciprocal, product $-1$). To partition a directed segment from $A$ to $B$ in ratio $m:n$, move $\frac{m}{m + n}$ of the way from $A$. Write a line with point-slope form $y - y_1 = m(x - x_1)$.

Volume and solids

A prism or cylinder is $Bh$; a pyramid or cone is one-third of that; a sphere is $\frac{4}{3}\pi r^3$. A cross section is the 2D shape from slicing a solid; a solid of revolution is formed by spinning a region (a rectangle about a side gives a cylinder). Density is amount per volume, so amount equals density times volume.

How this strand is examined

• Part I (2 credits). A dilation length, a trig ratio choice, an inscribed angle, a perpendicular slope, or a cone volume.
• Part II (2 credits). A partition point, a circle equation read-off, or an arc length. Show the key step.
• Part III and IV (4 to 6 credits). A similar-triangles modeling task, a trig word problem with elevation, a circle problem combining angles and segments, or a volume-and-density model. Show the setup and round as asked.

Check your knowledge

Work these as you would for credit.

1. A dilation with scale factor 4 maps a side of length 7 to what length? (1 credit)
2. In a right triangle, the side opposite a 40-degree angle is unknown and the hypotenuse is 15. Which ratio finds it, and what is its value? (2 credits)
3. An inscribed angle intercepts a 130-degree arc. Find the angle. (1 credit)
4. State the center and radius of $(x + 5)^2 + (y - 1)^2 = 49$. (2 credits)
5. Find the slope of a line perpendicular to the line through $(2, 1)$ and $(6, 9)$. (2 credits)
6. Partition the directed segment from $A(0, 0)$ to $B(9, 6)$ in ratio $2:1$. (2 credits)
7. Find the volume of a sphere of radius 3, in terms of $\pi$. (2 credits)
8. A cylinder has radius 5 m, height 4 m, and holds a liquid of density 800 kg per cubic meter. Find the mass when full, to the nearest thousand kg. (4 credits)