DC CAPE Geometry Expressing Geometric Properties with Equations Practice

Expressing Geometric Properties with Equations skills

translate between the geometric description and the equation for a conic section

use coordinates to prove simple geometric theorems algebraically

proving and applying theorems about lines, angles, triangles, and parallelograms with coordinates

Standards basis

Common Core State Standards (CCSS-M) — District of Columbia

How the DC CAPE reports it

DC CAPE math continues DC’s PARCC-era alignment to the Common Core State Standards and uses the same general development/review approach described by OSSE. The page says DC Math follows the same rigorous development and review process and measures the knowledge and skills that matter most for DC students, while also explicitly tying the assessments to CCSS.

Blueprint & weighting

DC CAPE uses the PARCC-derived / Common Core-based framework (by grade domains structure).

Official sources

OSSE DC CAPE
OSSE DC CAPE page content on CCSS and test specifications

8 free questions · 0/0 correct

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

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 $\dfrac{3}{4}\textsf{.}$

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

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 $x^2 + y^2 = 10^2 = 100\textsf{.}$