Grade 10 Math MCAS: a complete guide to the Geometry category

What this category demands

The Geometry category on the Grade 10 MCAS is broad, drawn from the G-CO, G-SRT, G-C, G-GPE, and G-GMD standards of the 2017 framework. It treats congruence and similarity through transformations, solves right triangles with the Pythagorean theorem and trigonometry, measures circles and solids, and uses the coordinate plane to classify figures. Many formulas live on the reference sheet, but the slope and distance formulas do not, and the reasoning behind congruence and similarity is tested directly. This guide ties together the dot-point pages, each with its own practice: congruence and rigid motions, similarity and dilations, right triangle trigonometry, circles, angles, and arcs, coordinate geometry, and volume and surface area.

Congruence through rigid motions

A rigid motion (translation, reflection, rotation) preserves lengths and angles. Two figures are congruent when a sequence of rigid motions maps one onto the other. Coordinate rules to know: reflection across the x-axis is $(x, y) \to (x, -y)$, across the y-axis is $(x, y) \to (-x, y)$, a translation adds to the coordinates, and a $180^\circ$ rotation about the origin is $(x, y) \to (-x, -y)$.

Similarity and dilations

A dilation scales a figure by a factor $k$ from a center, multiplying lengths by $k$ and keeping angles. Similar figures have equal angles and proportional sides; triangles are similar by angle-angle (AA). The key scaling fact: perimeter scales by $k$, area by $k^2$, and volume by $k^3$.

Right triangle trigonometry

The Pythagorean theorem $a^2 + b^2 = c^2$ (on the sheet) finds a third side from two sides. The trig ratios (on the sheet) relate an angle to two sides: $\sin = \frac{\text{opp}}{\text{hyp}}$, $\cos = \frac{\text{adj}}{\text{hyp}}$, $\tan = \frac{\text{opp}}{\text{adj}}$ (SOH-CAH-TOA). To find an angle from two sides, use the inverse trig function.

Circles, angles, and arcs

Circumference is $C = 2\pi r$ and area is $A = \pi r^2$ (both on the sheet). An arc or sector is the fraction $\frac{\theta}{360}$ of the circle: arc length $= \frac{\theta}{360}(2\pi r)$, sector area $= \frac{\theta}{360}(\pi r^2)$. A central angle equals its arc; an inscribed angle is half its intercepted arc, and an angle inscribed in a semicircle is $90^\circ$.

Coordinate geometry

The distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ and the midpoint formula $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ are core tools; the distance formula is not on the sheet. Slope tests relationships: equal slopes are parallel, negative-reciprocal slopes (product $-1$) are perpendicular. Together they classify triangles and quadrilaterals.

Volume and surface area

Volume formulas on the sheet: prism $Bh$, cylinder $\pi r^2 h$, sphere $\frac{4}{3}\pi r^3$, cone $\frac{1}{3}\pi r^2 h$, pyramid $\frac{1}{3}Bh$. Cones and pyramids carry the $\frac{1}{3}$. Volume is cubic units (capacity); surface area is square units (covering). Square the radius before multiplying, and keep $\pi$ exact unless a decimal is asked.

How this category is examined

• Selected-response (1 point, mixed calculator). A coordinate distance, a circle area, a transformation image, a volume. Technology-enhanced items may have you plot or drag.
• Short-answer (1 point). A single arc length, a missing side by Pythagoras, a midpoint.
• Constructed-response (multi-point). A trig problem with reasoning, a full coordinate classification, or a composite-solid volume with units.

Check your knowledge

Work these as you would for credit.

1. Reflect $(3, -5)$ across the y-axis. (1 point)
2. Triangles are similar with scale factor 4; a side of 6 corresponds to what length? (1 point)
3. A right triangle has legs 9 and 12. Find the hypotenuse. (2 points)
4. A circle has radius 8. Find its area in terms of $\pi$. (1 point)
5. Find the arc length of a $45^\circ$ sector in a circle of radius 12, in terms of $\pi$. (2 points)
6. Find the distance between $(2, 1)$ and $(7, 13)$. (2 points)
7. A cylinder has radius 5 and height 4. Find its volume in terms of $\pi$. (2 points)
8. A cone has radius 6 and height 10. Find its volume in terms of $\pi$. (2 points)