2025 (Aug.): NY Regents - Geometry

In quadrilateral $ABCD$ below, side $\overline{CD}$ is extended through $D$ to point $E$ such that $\overline{AFD}$ and $\overline{BFE}$ bisect each other, and $\overline{DE} \cong \overline{DC}$.

Prove $ABCD$ is a parallelogram.

An equilateral triangle is continuously rotated around one of its altitudes. The three-dimensional object formed is a

On the set of axes below, quadrilateral BDGF is rotated 90 degrees clockwise about the origin and then reflected over the y-axis. The image of quadrilateral BDGF is quadrilateral MQSP.

Side $\overline{BD}$ will always map onto

In right triangle $JOE$, hypotenuse $JO$ = 31.8 and $m\angle J$ = 38°. To the nearest tenth, the length of $\overline{EJ}$ is

The hemisphere below has a radius of 8 cm.

To the nearest cubic centimeter, the volume of the hemisphere is

In parallelogram $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Which information is sufficient to prove $ABCD$ is a rhombus?

Trapezoid $JOSH$, shown below, has non-parallel sides $\overline{JH}$ and $\overline{OS}$, $m\angle J = 65^\circ$, $m\angle O = 30^\circ$, $m\angle OSA = 80^\circ$, and $m\angle SHU = 60^\circ$.

What is $m\angle HSA$?

In $\triangle ABC$ below, points D and E are on $\overline{AB}$ and $\overline{CB}$, respectively, such that $\overline{DE} \parallel \overline{AC}$.

If $AD = 8$, $DB = 4$, and $DE = 6$, what is the length of $\overline{AC}$?

On the set of axes below, circle $C$ has a center with coordinates $(2, -1)$.

Which equation represents circle $C$?

On the set of axes below, $\triangle D'E'F'$ is the image of $\triangle DEF$.

A transformation that maps $\triangle DEF$ onto $\triangle D'E'F'$ is a

In circle $O$ below, secants $\overline{PCA}$ and $\overline{PDB}$ are drawn from external point $P$.

If $PA = 17$, $PD = 10$, and $BD = 12$, what is the length of $\overline{PC}$, to the nearest tenth?

In the diagram below, $\overline{CD} \parallel \overline{AB}$, and $\overline{CB}$ bisects $\angle ABD$.

Which statement must be true?

Line $h$ is represented by the equation $y = \dfrac{2}{3}x - 4$.
Which equation represents the line that is perpendicular to line $h$ and passes through the point $(6,1)$?

A wooden toy block can be modeled by a pyramid with a square base, as shown below. The height of the block is 17.4 cm and the square base has a side length of 8.2 cm.

The block is made of solid oak, which has a density of 0.77 $\text{g/cm}^3$. What is the mass of the block, to the nearest gram?

In $\triangle ABC$ below, midsegment $\overline{DE}$ is drawn.

If $DE = x + 3$ and $AC = 3x - 5$, what is the length of $\overline{DE}$?

Triangle $DUG$ is an isosceles right triangle with the right angle at $G$. If $DU = 10\sqrt{2}$, what is the length of $\overline{GU}$?

In $\triangle RST$ below, $RS = 9$ cm, $RT = 8$ cm, and $m\angle TRS = 55^\circ$.

What is the area of $\triangle RST$, to the nearest square centimeter?

Triangle $ABC$ is dilated by a scale factor of $2$ to map onto its image, $\triangle RST$, on the set of axes below.

What are the coordinates of the center of this dilation?

What is the perimeter of $\triangle ABC$, where the vertices have coordinates $A(-2,3)$, $B(-2,-1)$, and $C(6,-1)$?

In the diagram below, $\overline{GT}$ and $\overline{PF}$ intersect at E, and $\angle P \cong \angle F$.

Which equation is always true?

A section of sidewalk in the shape of a rectangular prism is being replaced. The sidewalk is 10 feet long, 4 feet wide, and 4 inches deep. A brand of concrete mix yields 0.6 cubic foot of concrete per bag. What is the minimum number of bags of concrete mix that must be purchased to completely replace this sidewalk?

The line $4x - 6y = 24$ is transformed by a dilation of scale factor 3 centered at the origin. Which equation represents the image of the line after this dilation?

A rhombus is graphed on the set of axes below.

Which transformation does not carry the rhombus onto itself?

In right triangle $HAY$ below, altitude $\overline{AL}$ is drawn to hypotenuse $\overline{HY}$.

If $HY = 25$ and $YA = 20$, the length of $\overline{AL}$ is

Square $ABCD$ has an area of 36. If the square is dilated by a scale factor of $\dfrac{1}{2}$ centered at $A$, what is the area of its image?

The endpoints of $\overline{CAS}$ are C($-3,1$) and S($7,6$). Determine and state the coordinates of point A such that the ratio of $CA:AS$ is $3:2$.

The ramp shown in the diagram below has an angle of elevation of $4.8^\circ$. The ramp is built to a landing $0.6$ m above the ground.

Determine and state the length of the ramp, to the nearest tenth of a meter.

Angle $KML$ is the vertex angle of isosceles triangle $KLM$ below. Side $\overline{LM}$ is extended through vertex $M$ to point $N$.

If $m\angle K = 15^\circ$, determine and state $m\angle KMN$.

In the diagram below of circle $L$, the area of the shaded sector $KLM$ is $7.5\pi$ and $LK = 5$.

Determine and state the degree measure of angle $KLM$, the central angle of the shaded sector.

Using a compass and straightedge, construct the image of point A after a reflection over $\overline{BC}$.
[Leave all construction marks.]

Suggestion: Utilize a separate sheet of paper to answer this question and submit a photo of it in the Drawing space.