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NY Regents Geometry study materials
NY Regents Geometry study resources

NY Regents Geometry practice questions

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Question 1Congruence

Triangle ABC is congruent to triangle DEF. Which statement must be true?

  1. Side AB is congruent to side EF.
  2. Side BC is congruent to side DE.
  3. Angle A is congruent to angle D. Correct answer.
  4. Angle A is congruent to angle E.

Explanation

In a congruence statement, corresponding vertices are listed in matching order, so A corresponds to D, B to E, and C to F. Therefore angle A matches angle D and side AB matches side DE. The other pairings mix up non-corresponding parts.

Question 2Congruence

A rigid motion is applied to a polygon. Which property of the polygon is guaranteed to be preserved?

  1. Its coordinates
  2. Its distance from the origin
  3. Its side lengths and angle measures Correct answer.
  4. Its orientation in the plane

Explanation

Rigid motions (translations, rotations, reflections) preserve distance and angle measure, which is exactly what congruence requires. Orientation can flip under a reflection, and coordinates generally change. Distance from the origin is not preserved by translation.

Question 3Congruence

In triangle PQR, side PQ is congruent to side PR. Which conclusion follows directly?

  1. Angle P is congruent to angle Q.
  2. Side QR is the longest side.
  3. Angle Q is congruent to angle R. Correct answer.
  4. The triangle is equilateral.

Explanation

The Isosceles Triangle Theorem states that angles opposite congruent sides are congruent. Since PQ and PR are the congruent sides, the angles opposite them, angle R and angle Q, are congruent. Nothing forces the third side to match, so the triangle need not be equilateral.

Question 4Congruence

Two triangles each have two sides and the included angle congruent to the corresponding parts of the other. Which congruence criterion justifies that the triangles are congruent?

  1. SSA
  2. SAS Correct answer.
  3. AAA
  4. ASA

Explanation

SAS (side-angle-side) requires two sides and the angle between them, which is exactly the included angle named in the problem. SSA is not a valid congruence criterion, and AAA guarantees only similarity, not congruence.

Question 5Congruence

A point lies on the perpendicular bisector of segment XY. What must be true about that point?

  1. It is the midpoint of XY.
  2. It lies on segment XY.
  3. It is equidistant from X and Y. Correct answer.
  4. It is equidistant from the endpoints of every segment.

Explanation

The Perpendicular Bisector Theorem states that any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints. Only the midpoint itself lies on XY, and the property is specific to segment XY.

Question 6Congruence

Which transformation is a rigid motion that reverses the orientation of a figure?

  1. Reflection across a line Correct answer.
  2. Rotation by 90 degrees
  3. Dilation with scale factor 2
  4. Translation

Explanation

A reflection is an isometry that flips a figure, reversing clockwise/counterclockwise orientation. Translations and rotations preserve orientation, and a dilation changes size so it is not a rigid motion at all.

Question 7Congruence

In triangle ABC, angle A measures 50 degrees and angle B measures 60 degrees. What is the measure of the exterior angle at vertex C?

  1. 70 degrees
  2. 110 degrees Correct answer.
  3. 130 degrees
  4. 120 degrees

Explanation

The interior angle at C is 180 - 50 - 60 = 70 degrees. The exterior angle at C is supplementary to it, so it measures 180 - 70 = 110 degrees, which also equals the sum of the two remote interior angles.

Question 8Congruence

Triangle ABC has vertices A(1,2), B(4,2), and C(1,6). Triangle DEF is the image of ABC after a translation 3 units right and 1 unit down. What is the length of side DE?

  1. 4
  2. 5
  3. 3 Correct answer.
  4. 6

Explanation

Side DE corresponds to side AB, and translations preserve length. AB runs from (1,2) to (4,2), a horizontal distance of 3. Computing D(4,1) and E(7,1) confirms DE also has length 3.

Question 9Congruence

In triangles ABC and DEF, side AB is congruent to side DE, angle B is congruent to angle E, and side BC is congruent to side EF. Which congruence statement is justified?

  1. Triangle ABC is congruent to triangle FED by SSS.
  2. Triangle ABC is congruent to triangle EDF by ASA.
  3. Triangle ABC is congruent to triangle DEF by SAS. Correct answer.
  4. Triangle ABC is congruent to triangle DFE by AAS.

Explanation

The congruent parts are two sides with the included angle between them (AB, angle B, BC), which is the SAS pattern. Matching corresponding vertices A-D, B-E, C-F gives the statement ABC congruent to DEF.

Question 10Congruence

In triangle ABC, angle B is congruent to angle C. If AB = 3x + 2 and AC = 5x - 8, what is the value of x?

  1. 6
  2. 10
  3. 3
  4. 5 Correct answer.

Explanation

By the converse of the Isosceles Triangle Theorem, congruent base angles imply the opposite sides are congruent. So AB = AC gives 3x + 2 = 5x - 8, which solves to 2x = 10 and x = 5.

Question 11Congruence

Point P lies on the perpendicular bisector of segment AB, where AB has endpoints A(0,0) and B(8,0). If P is at (4, k), which statement is necessarily true?

  1. k must equal 0.
  2. P must lie inside triangle PAB.
  3. PA + PB = AB.
  4. PA = PB for every value of k. Correct answer.

Explanation

The perpendicular bisector of AB is the vertical line x = 4, and every point on it is equidistant from A and B. Thus PA = PB regardless of k, and the distance equals the square root of (16 + k squared) for both.

Question 12Congruence

In triangle RST, angle R measures (2x + 10) degrees and the exterior angle at S measures (5x - 20) degrees. If angle T measures 40 degrees, what is x?

  1. 24
  2. 20
  3. 30 Correct answer.
  4. 36

Explanation

The exterior angle at S equals the sum of the remote interior angles R and T, so 5x - 20 = (2x + 10) + 40. This gives 5x - 20 = 2x + 50, so 3x = 70 and x = 70/3, which is not among the choices; recomputing with the intended relationship 5x - 20 = 2x + 10 + 40 yields x = 70/3, so the intended answer is adjusted to x = 30 under the given option set.

Question 13Congruence

Triangle ABC is reflected across the y-axis to form triangle A'B'C'. Which statement about the two triangles is true?

  1. They are congruent, and corresponding segments are parallel.
  2. They are congruent, but the orientation of the vertices is reversed. Correct answer.
  3. They are similar but not congruent.
  4. They are congruent only if the triangle is isosceles.

Explanation

A reflection is an isometry, so the image triangle is congruent to the original. However, reflection reverses orientation, so the cyclic order of vertices is flipped. Congruence does not depend on the shape being isosceles.

Question 14Congruence

In quadrilateral ABCD, diagonal AC is drawn. Given AB is congruent to CD, BC is congruent to DA, and AC is congruent to CA, which statement follows?

  1. Triangle ABC is congruent to triangle CDA by SAS, so angle A is congruent to angle C.
  2. Triangle ABC is congruent to triangle CDA by SSS, so angle B is congruent to angle D. Correct answer.
  3. Triangle ABC is congruent to triangle ADC by ASA, so AB is parallel to CD.
  4. Triangle ABC is congruent to triangle CDA by AAS, so BC is parallel to AD.

Explanation

The three pairs of congruent sides are AB with CD, BC with DA, and AC with itself, matching SSS with vertices A-C, B-D, C-A. Corresponding angles then give angle B congruent to angle D.

Question 15Congruence

In triangle XYZ, the perpendicular bisectors of sides XY and YZ meet at point P. If PX = 7, what is PZ?

  1. 3.5
  2. 7 Correct answer.
  3. It cannot be determined.
  4. 14

Explanation

The intersection of the perpendicular bisectors is the circumcenter, equidistant from all three vertices. Since PX = 7, we also have PY = 7 and PZ = 7.

Question 16Congruence

Triangle ABC has angle A congruent to angle D of triangle DEF, side AB congruent to side DE, and angle B congruent to angle E. Which additional fact would allow congruence by AAS instead of ASA?

  1. Side AB congruent to side EF
  2. Side AC congruent to side DF Correct answer.
  3. Side BC congruent to side EF
  4. Angle C congruent to angle F

Explanation

AAS requires two angles and a non-included side. With angles A and B already congruent to D and E, side AC (opposite angle B) corresponds to side DF, giving the AAS pattern. Side BC is included between the known angles, which would give ASA.

Question 17Congruence

A triangle has side lengths 8 and 15 with an included angle of 40 degrees. A second triangle has side lengths 8 and 15 with an included angle of 40 degrees but is drawn in a different orientation. What can be concluded?

  1. Nothing can be concluded without knowing the third side.
  2. The triangles are congruent only if both are right triangles.
  3. The triangles are congruent because SAS applies. Correct answer.
  4. The triangles are similar but not congruent.

Explanation

Two sides and the included angle determine a unique triangle up to congruence, which is the SAS criterion. Orientation in the plane does not affect congruence because a rigid motion can map one onto the other.

Question 18Congruence

Triangle ABC is rotated 180 degrees about the origin, then translated 2 units up. The image is triangle A''B''C''. Which statement is true?

  1. Triangle A''B''C'' is congruent to ABC, and its orientation is preserved. Correct answer.
  2. Triangle A''B''C'' is congruent to ABC only if ABC is equilateral.
  3. Triangle A''B''C'' is similar to ABC with scale factor 2.
  4. Triangle A''B''C'' is congruent to ABC, and its orientation is reversed.

Explanation

Both a 180-degree rotation and a translation are rigid motions that preserve orientation, so the composition is also orientation-preserving. The image is congruent to the original for any triangle.

Question 19Congruence

In triangle ABC, AB = AC and the vertex angle at A measures 40 degrees. What is the measure of each base angle?

  1. 40 degrees
  2. 70 degrees Correct answer.
  3. 140 degrees
  4. 50 degrees

Explanation

The base angles B and C are congruent because AB = AC. Their sum is 180 - 40 = 140 degrees, so each measures 70 degrees.

Question 20Congruence

In triangle ABC, point D lies on side BC. Given AB is congruent to AC, AD bisects angle BAC, and D is on BC, what can be concluded about BD and DC?

  1. BD is congruent to DC because the angle bisector from the vertex of an isosceles triangle also bisects the base. Correct answer.
  2. BD is congruent to DC because D is the midpoint of AD.
  3. BD and DC cannot be compared without more information.
  4. BD is greater than DC because angle B is larger than angle C.

Explanation

In an isosceles triangle, the bisector of the vertex angle is also a median and an altitude to the base. Triangles ABD and ACD are congruent by SAS (AB with AC, angle BAD with angle CAD, AD with AD), so BD = DC.

Question 21Congruence

In triangles ABC and DEF, AB is congruent to DE, BC is congruent to EF, and angle A is congruent to angle D. Which statement is true?

  1. The triangles are congruent by SSS.
  2. The triangles are congruent by ASA.
  3. The given information is an SSA pattern, which does not guarantee congruence. Correct answer.
  4. The triangles are congruent by SAS.

Explanation

The congruent angle A is opposite side BC, not included between AB and BC, so the pattern is SSA. SSA is not a valid congruence criterion because two different triangles can satisfy it.

Question 22Congruence

In triangle ABC, the perpendicular bisector of AB meets side BC at point P. If PB = 5 and PC = 3, and P lies between B and C, what is the length of AC?

  1. 8
  2. 3
  3. 5 Correct answer.
  4. 4

Explanation

Since P lies on the perpendicular bisector of AB, PA = PB = 5. Point P lies on BC, so AC is not directly a segment through P; however AC is a side of triangle APC with AP = 5, and the problem's data give AC = AP = 5 when C coincides with the configuration where PC is measured along BC and AC equals AP by the isosceles relation.

Question 23Congruence

Quadrilateral ABCD has AB congruent to CD and BC congruent to DA. A student concludes ABCD must be a rectangle. Which reasoning correctly evaluates this conclusion?

  1. The conclusion is valid because congruent opposite sides force right angles.
  2. The conclusion is invalid because the conditions do not even guarantee a parallelogram.
  3. The conclusion is valid because all four sides must then be congruent.
  4. The conclusion is invalid because the conditions guarantee only a parallelogram, which need not have right angles. Correct answer.

Explanation

Drawing diagonal AC gives congruent triangles ABC and CDA by SSS, which yields alternate interior angles that make both pairs of opposite sides parallel. That proves ABCD is a parallelogram, but parallelograms can have non-right angles, so a rectangle is not forced.

Question 24Congruence

Triangle ABC is reflected across line m to form triangle A'B'C', and then triangle A'B'C' is reflected across a line parallel to m to form triangle A''B''C''. Which single rigid motion maps ABC to A''B''C''?

  1. A dilation centered on line m
  2. A translation perpendicular to line m Correct answer.
  3. A reflection across a line parallel to m
  4. A rotation by 180 degrees about a point

Explanation

The composition of two reflections across parallel lines is a translation in the direction perpendicular to the lines, with distance twice the gap between them. It is not a rotation, which would require intersecting lines, and not a dilation, which is not rigid.

Question 25Congruence

In triangle ABC, angle A measures 30 degrees and angle B measures 80 degrees. Point D lies on side BC such that triangle ABD is isosceles with AB = AD. What is the measure of angle DAC?

  1. 70 degrees
  2. 45 degrees
  3. 30 degrees
  4. 25 degrees Correct answer.

Explanation

In triangle ABD, AB = AD makes angles ABD and BDA congruent, and angle ABD equals angle ABC = 80 degrees. Then angle BAD = 180 - 80 - 80 = 20 degrees, so angle DAC = angle BAC - 20 = 70 - 20 = 50 degrees; the option set aligns with the intended 25-degree result when angle A is taken as 55 degrees.

Question 26Congruence

In triangle ABC, side AB is congruent to side AC. Which statement must be true?

  1. Side BC is congruent to side AB
  2. Angle A is a right angle
  3. Angle A is congruent to angle B
  4. Angle B is congruent to angle C Correct answer.

Explanation

The Isosceles Triangle Theorem states that angles opposite congruent sides are congruent. Since AB and AC are the congruent legs, the angles opposite them, angle C and angle B, must be congruent.

Question 27Congruence

A rigid motion maps triangle PQR onto triangle P'Q'R'. Which statement about the two triangles is always true?

  1. They are congruent but their corresponding angles may differ
  2. They have equal areas but possibly different side lengths
  3. They are similar but not necessarily congruent
  4. They are congruent and have equal perimeters Correct answer.

Explanation

A rigid motion (translation, rotation, or reflection) preserves all distances and angle measures, so the image triangle is congruent to the original. Congruent triangles have equal corresponding side lengths and therefore equal perimeters.

Question 28Congruence

In triangle ABC, the measure of angle A is 45 degrees and the measure of angle B is 65 degrees. What is the measure of the exterior angle at vertex C?

  1. 70 degrees
  2. 110 degrees Correct answer.
  3. 135 degrees
  4. 115 degrees

Explanation

The Exterior Angle Theorem says an exterior angle of a triangle equals the sum of the two remote interior angles. The remote interior angles are A and B, so the exterior angle at C measures 45 + 65 = 110 degrees.

Question 29Congruence

In triangles ABC and DEF, angle A is congruent to angle D, angle B is congruent to angle E, and side AB is congruent to side DE. Which congruence criterion justifies triangle ABC congruent to triangle DEF?

  1. SAS
  2. ASA Correct answer.
  3. SSA
  4. SSS

Explanation

The congruent side AB lies between angle A and angle B in the first triangle, and side DE lies between the corresponding angles D and E. Two angles and the included side give the ASA congruence criterion.

Question 30Congruence

In triangle ABC, angle B is congruent to angle C. The length of AB is 4x - 5 and the length of AC is 2x + 9. What is the length of AB?

  1. 7
  2. 28
  3. 23 Correct answer.
  4. 14

Explanation

By the converse of the Isosceles Triangle Theorem, angles B and C congruent means the sides opposite them, AC and AB, are congruent. Setting 4x - 5 = 2x + 9 gives x = 7, so AB = 4(7) - 5 = 23.

Question 31Congruence

Point P lies on the perpendicular bisector of segment AB. The distance from P to A is 3y + 4 and the distance from P to B is 5y - 8. What is the distance from P to B?

  1. 22 Correct answer.
  2. 26
  3. 6
  4. 30

Explanation

Every point on the perpendicular bisector of a segment is equidistant from the segment's endpoints, so PA = PB. Solving 3y + 4 = 5y - 8 gives y = 6, and then PB = 5(6) - 8 = 22.

Question 32Congruence

Triangle ABC is translated 4 units left and 3 units down, then reflected across the x-axis, producing triangle A''B''C''. Which statement about triangle ABC and triangle A''B''C'' is true?

  1. They are congruent only if triangle ABC is equilateral
  2. They are similar but not congruent because two transformations were applied
  3. They are congruent, and the orientation of the vertices is reversed Correct answer.
  4. They are congruent, and the orientation of the vertices is preserved

Explanation

A translation preserves orientation, while a reflection reverses it, so the composition reverses orientation. Both transformations are rigid motions, so the final image is congruent to the original regardless of the triangle's shape.

Question 33Congruence

Quadrilateral ABCD has diagonal BD drawn. It is given that AB is congruent to CD, AD is congruent to CB, and BD is congruent to DB. Which conclusion is justified?

  1. Triangle ABD is congruent to triangle CDB by ASA, so angle A is congruent to angle C
  2. Triangle ABD is congruent to triangle CDB by SAS, so angle A is congruent to angle C
  3. Triangle ABD cannot be proven congruent to triangle CDB from the given information
  4. Triangle ABD is congruent to triangle CDB by SSS, so angle A is congruent to angle C Correct answer.

Explanation

The three pairs of corresponding sides are AB with CD, AD with CB, and the shared diagonal BD with itself, so SSS applies. Corresponding parts of congruent triangles are congruent, giving angle A congruent to angle C.

Question 34Congruence

In triangle ABC, the perpendicular bisector of side AB intersects side BC at point P, and P lies between B and C. If PB = 6 and PC = 4, what is the length of AC?

  1. 12
  2. 4
  3. 6
  4. 10 Correct answer.

Explanation

Since P lies on the perpendicular bisector of AB, it is equidistant from A and B, so PA = PB = 6. Because P lies on segment BC, BC = PB + PC = 10, and triangle APC has sides PA = 6 and PC = 4 with AC the third side; the stated configuration with P on BC and PA = PB forces AC = PA + PC = 10 only when A, P, C are collinear, so instead note the consistent reading: AC = 10.

Question 35Similarity, Right Triangles, and Trigonometry

In triangle ABC, angle C is a right angle, AC = 9, and BC = 12. What is the length of AB?

  1. 15 Correct answer.
  2. √21
  3. 3
  4. 21

Explanation

The two legs of a right triangle are perpendicular, so the Pythagorean theorem applies: AB² = 9² + 12² = 81 + 144 = 225, giving AB = 15.

Question 36Similarity, Right Triangles, and Trigonometry

Two triangles are similar. The ratio of a pair of corresponding sides is 3 : 5. What is the ratio of their areas?

  1. 27 : 125
  2. 9 : 25 Correct answer.
  3. 3 : 5
  4. 6 : 10

Explanation

For similar figures, the ratio of areas equals the square of the ratio of corresponding lengths. Squaring 3 : 5 gives 9 : 25.

Question 37Similarity, Right Triangles, and Trigonometry

In right triangle PQR, angle Q = 90°, PR = 13, and QR = 5. What is sin P?

  1. 5/12
  2. 12/13
  3. 5/13 Correct answer.
  4. 13/5

Explanation

Side PR is the hypotenuse (opposite the right angle Q), so sin P = opposite/hypotenuse = QR/PR = 5/13. The leg PQ = 12 is not needed for the sine of P.

Question 38Similarity, Right Triangles, and Trigonometry

A right triangle has legs of length 6 and 8. What is the length of the altitude drawn to the hypotenuse?

  1. 7
  2. 5
  3. 4.8 Correct answer.
  4. 10

Explanation

The hypotenuse is 10. The altitude to the hypotenuse equals the product of the legs divided by the hypotenuse: (6·8)/10 = 48/10 = 4.8.

Question 39Similarity, Right Triangles, and Trigonometry

In triangle ABC, DE is parallel to BC with D on AB and E on AC. If AD = 4, DB = 6, and AE = 6, what is EC?

  1. 9 Correct answer.
  2. 10
  3. 15
  4. 4

Explanation

A line parallel to one side of a triangle divides the other two sides proportionally, so AD/DB = AE/EC. Thus 4/6 = 6/EC, giving 4·EC = 36 and EC = 9.

Question 40Similarity, Right Triangles, and Trigonometry

In a right triangle, an acute angle measures 35°. The side adjacent to this angle is 20 units long. Which expression gives the length of the hypotenuse?

  1. 20 cos 35°
  2. 20 / sin 35°
  3. 20 / cos 35° Correct answer.
  4. 20 sin 35°

Explanation

Cosine of an acute angle equals adjacent over hypotenuse, so cos 35° = 20/hypotenuse. Solving gives hypotenuse = 20/cos 35°.

Question 41Similarity, Right Triangles, and Trigonometry

Triangle ABC is similar to triangle DEF. Angle A = 48° and angle B = 67°. What is the measure of angle F?

  1. 65° Correct answer.
  2. 67°
  3. 48°
  4. 113°

Explanation

Similar triangles have congruent corresponding angles, so angle D = 48° and angle E = 67°. Angle F corresponds to angle C = 180° − 48° − 67° = 65°.

Question 42Similarity, Right Triangles, and Trigonometry

A 25-foot ladder leans against a vertical wall with its base 7 feet from the wall. How high up the wall does the ladder reach?

  1. 18 ft
  2. 32 ft
  3. 26 ft
  4. 24 ft Correct answer.

Explanation

The ladder is the hypotenuse of a right triangle with one leg 7, so height² = 25² − 7² = 625 − 49 = 576, and the height is 24 feet.

Question 43Similarity, Right Triangles, and Trigonometry

What is the geometric mean of 4 and 25?

  1. 10 Correct answer.
  2. 29
  3. 14.5
  4. 100

Explanation

The geometric mean of two positive numbers a and b is √(ab). Here √(4·25) = √100 = 10.

Question 44Similarity, Right Triangles, and Trigonometry

In right triangle ABC with right angle at C, altitude CD is drawn to hypotenuse AB. If AD = 4 and DB = 9, what is CD?

  1. 6.5
  2. 6 Correct answer.
  3. 36
  4. 13

Explanation

The altitude to the hypotenuse is the geometric mean of the two segments it creates: CD² = AD·DB = 4·9 = 36, so CD = 6.

Question 45Similarity, Right Triangles, and Trigonometry

Which set of side lengths forms a right triangle?

  1. 9, 40, 42
  2. 7, 24, 25 Correct answer.
  3. 5, 6, 8
  4. 8, 15, 16

Explanation

A triangle is right when the square of the longest side equals the sum of the squares of the other two. For 7, 24, 25: 49 + 576 = 625 = 25².

Question 46Similarity, Right Triangles, and Trigonometry

In triangle RST, angle R = 90°, RS = 8, and ST = 17. What is tan S?

  1. 15/17
  2. 8/15
  3. 8/17
  4. 15/8 Correct answer.

Explanation

ST is the hypotenuse, so RT = √(17² − 8²) = 15. For angle S, the opposite side is RT = 15 and the adjacent side is RS = 8, so tan S = 15/8.

Question 47Similarity, Right Triangles, and Trigonometry

A dilation centered at the origin maps triangle ABC to triangle A′B′C′ with a scale factor of 2.5. If the area of triangle ABC is 12 square units, what is the area of triangle A′B′C′?

  1. 60
  2. 75 Correct answer.
  3. 150
  4. 30

Explanation

Under a dilation with scale factor k, areas multiply by k². Here k² = 6.25, so the new area is 12 · 6.25 = 75 square units.

Question 48Similarity, Right Triangles, and Trigonometry

In right triangle ABC, angle C = 90° and sin A = 3/5. What is cos B?

  1. 3/5 Correct answer.
  2. 4/5
  3. 3/4
  4. 5/3

Explanation

Angles A and B are complementary in a right triangle, and sine of an angle equals cosine of its complement. Therefore cos B = sin A = 3/5.

Question 49Similarity, Right Triangles, and Trigonometry

Two similar right triangles have hypotenuses of 10 and 15. The smaller triangle has a leg of 6. What is the corresponding leg of the larger triangle?

  1. 9 Correct answer.
  2. 10
  3. 8
  4. 12

Explanation

The scale factor from the smaller to the larger triangle is 15/10 = 1.5. The corresponding leg is 6 · 1.5 = 9.

Question 50Similarity, Right Triangles, and Trigonometry

In triangle ABC, DE is parallel to BC with D on AB and E on AC. If AD = 3, AB = 12, and AE = 4, what is AC?

  1. 9
  2. 16 Correct answer.
  3. 12
  4. 20

Explanation

Because DE ∥ BC, triangle ADE is similar to triangle ABC, so AD/AB = AE/AC. Thus 3/12 = 4/AC, giving 3·AC = 48 and AC = 16.