MCAP Geometry Expressing Geometric Properties with Equations Practice

Expressing Geometric Properties with Equations skills

G.GPE.A Translate between the geometric description and the equation for a conic section.

G.GPE.B Use coordinates to prove simple geometric theorems algebraically.

Standards basis

Maryland College and Career Ready Standards for Mathematics (MCCRSM) — Maryland

How the MCAP reports it

MCAP is Maryland's statewide successor to PARCC, built on the Maryland College and Career Ready Standards for Mathematics (which mirror CCSS-M) — it is not a Smarter Balanced administration despite the claims-style family resemblance. Each grade/course High Level Blueprint (September 2022) organizes the assessment into three subclaims: Content (23-24 one-point machine-scored items, listed per domain/conceptual category as MCCRSM cluster headings), Reasoning (6 items), and Modeling (6 items), the latter two defined by per-grade evidence statements and including constructed-response items. Cluster headings are transcribed verbatim from the blueprints, including their minor typographical deviations from the parallel CCSS-M headings; two evident misprints are normalized (a duplicated sentence fragment on 4.OA.C, and the domain codes printed as G.P.E / T.TF for G.GPE / F.TF). Since March 2025, grade 6-7 students enrolled in a high-school mathematics course may take the corresponding MCAP course assessment instead of the grade-level test.

Blueprint & weighting

Each High Level Blueprint (September 2022) publishes per-domain operational item counts. Content Subclaim (1-point machine-scored items): Grade 3 — OA 7, NBT 2, NF 7, Measurement 5, Geometry 2 (23 items); Grade 4 — OA 4, NBT 5, NF 10, Measurement 3, Geometry 1 (23); Grade 5 — OA 2, NBT 6, NF 9, Measurement 4, Geometry 2 (23); Grade 6 — RP 3, NS 8, EE 8, Geometry 2, SP 2 (23); Grade 7 — RP 8, NS 4, EE 5, Geometry 3, SP 3 (23); Grade 8 — NS 2, EE 10, Functions 5, Geometry 4, SP 2 (23); Algebra I — Number and Quantity 1, Algebra 12, Functions 9, Statistics 2 (24); Geometry — G.CO 7, G.SRT 8, G.C 3, G.GPE 3, G.GMD 1, G.MG 1 (23); Algebra II — Number and Quantity 3, Algebra 8, Functions 11, Statistics 1 (23). Every assessment adds 6 Reasoning Subclaim and 6 Modeling Subclaim operational items — four 1-point machine-scored each, plus two constructed-response items (two 3-point in grades 3-4; one 3-point and one 4-point in grades 5-8; two 4-point in Algebra I, Geometry, and Algebra II) — so a form carries 35 operational items (36 for Algebra I).

Official sources

MSDE — MCAP Mathematics (current program page: grades 3-8, Algebra I, Geometry, Algebra II)
MSDE — MCAP Overview / High Level Blueprints summary
MSDE — MCAP Grade 3 Mathematics High Level Blueprint (September 2022)

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Practice playeasy

A circle in the coordinate plane has a diameter with endpoints $(-6, -5)$ and $(-6, 9)\textsf{.}$ What is the radius of the circle?

The endpoints share the same $x$ -coordinate, so the diameter is vertical with length $|9 - (-5)| = 14\textsf{.}$ The radius is half the diameter, so $r = \dfrac{14}{2} = 7\textsf{.}$

Practice playeasy

The equation $x^2 + (y - 6)^2 = 64$ represents Circle A. Circle B is obtained by shifting Circle A down $6$ units in the coordinate plane. Which of the following equations represents Circle B?

Circle A has center $(0, 6)$ and radius $8$ . Shifting down $6$ moves the center to the origin, so Circle B is $x^2 + y^2 = 64$ .

Practice playeasy

The graph of the equation $(x - 4)^2 + (y + 5)^2 = 36$ is a circle in the coordinate plane. The point $(a, b)$ lies on the circle. Which of the following is a possible value for $a\textsf{?}$

The center is $(4, -5)$ and the radius is $6\textsf{.}$ A point on the circle must satisfy $4 - 6 \le a \le 4 + 6\textsf{,}$ so $-2 \le a \le 10\textsf{.}$ The value $3$ is in this interval.

Practice playeasy

Identify the conic: $\dfrac{x^{2}}{16} - \dfrac{y^{2}}{9} = 1$ .

One squared term is subtracted from the other: this is a hyperbola.

Practice playeasy

A circle in the coordinate plane is centered at the origin and passes through the point $(6, 8)\textsf{.}$ Which equation represents the circle?

The radius is the distance from $(0, 0)$ to $(6, 8)\textsf{:}$ $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$ units. Standard form is $(x - h)^2 + (y - k)^2 = r^2\textsf{,}$ so $x^2 + y^2 = 10^2 = 100\textsf{.}$

Practice playeasy

Find the midpoint of $(2, 8)$ and $(6, 4)$ . What is the $y$ -coordinate?

$y$ -coordinate $= \dfrac{8 + 4}{2} = \dfrac{12}{2} = 6$ .

Practice playmedium

The equation of a line is $y = -x + 7$ . Which equation represents the line that is parallel to this line and passes through the point $(4, -2)$ ?

Parallel lines have the same slope, so $m = -1$ . Substitute $(4, -2)$ into $y = -x + b$ : $-2 = -4 + b$ , so $b = 2$ . The equation is $y = -x + 2\textsf{.}$

Practice playeasy

Line $m$ is defined by $y = \dfrac{3}{4}x + 8$ . Line $n$ is parallel to line $m$ in the coordinate plane. What is the slope of line $n$ ?

Parallel lines have the same slope. In $y = \dfrac{3}{4}x + 8$ , the slope is $\dfrac{3}{4}$ , so the slope of line $n$ is also $\dfrac{3}{4}\textsf{.}$

MCAP Geometry Expressing Geometric Properties with Equations. Practice it free.

Expressing Geometric Properties with Equations skills

G.GPE.A Translate between the geometric description and the equation for a conic section.

G.GPE.B Use coordinates to prove simple geometric theorems algebraically.

Topics mapped to this domain

Circle diameter endpoints to radius (SAT)

Circle equation translations (ACT/SAT/PSAT 8/9/10)

Possible coordinates on a circle (SAT)

Conic sections

Equation of a circle

Midpoint formula

Parallel and perpendicular lines through a point (SAT)

Parallel and perpendicular slopes from slope-intercept form

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