ATLAS Geometry Expressing Geometric Properties with Equations Practice

Expressing Geometric Properties with Equations skills

Use coordinates to prove simple geometric theorems algebraically

Prove the slope criteria for parallel and perpendicular lines

Find the equation of a parabola given a focus and directrix

Find the equation of a circle given its center and radius

Standards basis

Arkansas Academic Standards for Mathematics (CCSS-M–aligned / Arkansas-adopted math standards) — Arkansas

How the ATLAS reports it

Arkansas’s statewide assessment system is ATLAS, which the ADE page says began serving in spring 2024. The official public page does not publish a math blueprint there, so the most defensible coverage map is the Arkansas math standards structure that ATLAS is aligned to.

Blueprint & weighting

ATLAS uses the Independent / state-specific framework (by grade domains structure).

Official sources

Arkansas DESE Assessment: Test Scores
Arkansas DESE home page
Arkansas DESE: 3–10 ATLAS Content Assessments

8 free questions · 0/0 correct

Practice playeasy

A circle in the coordinate plane has a diameter with endpoints $(2.5, -1)$ and $(2.5, 7)\textsf{.}$ What is the radius of the circle?

The endpoints share the same $x$ -coordinate, so the diameter is vertical with length $|7 - (-1)| = 8\textsf{.}$ The radius is half the diameter, so $r = \dfrac{8}{2} = 4\textsf{.}$

Practice playmedium

The equation $(x - 1)^2 + (y + 4)^2 = 49$ represents Circle A. Circle B is obtained by shifting Circle A up $6$ units in the coordinate plane. Which of the following equations represents Circle B?

Circle A has center $(1, -4)$ and radius $7$ . Shifting up $6$ moves the center to $(1, 2)$ , so Circle B is $(x - 1)^2 + (y - 2)^2 = 49$ .

Practice playeasy

A circle in the coordinate plane has equation $(x + 1)^{2} + (y - 7)^{2} = 4\textsf{.}$ What is the $y$ -coordinate of the center of the circle?

$(y - 7)^{2}$ means $k = 7\textsf{,}$ so the center is $(-1, 7)\textsf{.}$

Practice playmedium

The graph of the equation $(x + 1)^2 + (y + 10)^2 = 144$ 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 $(-1, -10)$ and the radius is $12\textsf{.}$ A point on the circle must satisfy $-1 - 12 \le a \le -1 + 12\textsf{,}$ so $-13 \le a \le 11\textsf{.}$ The value $6$ is in this interval.

Practice playmedium

The equation of a line is $y = \dfrac{1}{3}x + 4$ . Which equation represents the line that is perpendicular to this line and passes through the point $(3, 2)$ ?

Perpendicular lines have slopes that are negative reciprocals, so $m = -3$ . Substitute $(3, 2)$ into $y = -3x + b$ : $2 = -3(3) + b = -9 + b$ , so $b = 11$ . The equation is $y = -3x + 11\textsf{.}$

Practice playeasy

Line $u$ is defined by $y = \dfrac{2}{5}x - 3$ . Line $v$ is perpendicular to line $u$ in the coordinate plane. What is the slope of line $v$ ?

Perpendicular lines have slopes that are negative reciprocals. The slope of line $u$ is $\dfrac{2}{5}$ , so the slope of line $v$ is $-\dfrac{5}{2}\textsf{.}$

Practice playmedium

A circle in the coordinate plane has a diameter with endpoints $(0, 0)$ and $(6, 8)\textsf{.}$ What is the radius of the circle?

The diameter length is $\sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\textsf{.}$ The radius is half the diameter, so $r = \dfrac{10}{2} = 5\textsf{.}$

Practice playmedium

The equation $(x + 5)^2 + (y - 3)^2 = 25$ represents Circle A. Circle B is obtained by shifting Circle A right $2$ units in the coordinate plane. Which of the following equations represents Circle B?