ISEE Upper Level Quantitative Reasoning Quiz: Congruence And Similarity
20 questions · exam conditions
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Congruence And SimilarityQuestion 1 of 20

Triangle JKL is congruent to triangle MNO by the SAS postulate. If JK = MN = 8, angle K = angle N = 72°, and KL = NO = 5, what additional information would be sufficient to prove the triangles are congruent by SSS instead?

JL = MO and all angles are equal
JL = MO, which can be calculated using the Law of Cosines
Angle J = angle M = 180° - 72° - angle L
The triangles have equal areas and perimeters
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ISEE Upper Level Quantitative Reasoning Quiz

ISEE Upper Level Quantitative Reasoning Quiz: Congruence And Similarity

Practice Congruence And Similarity in ISEE Upper Level Quantitative Reasoning with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Congruence And Similarity, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level Quantitative Reasoning.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Triangle JKL is congruent to triangle MNO by the SAS postulate. If JK = MN = 8, angle K = angle N = 72°, and KL = NO = 5, what additional information would be sufficient to prove the triangles are congruent by SSS instead?

  1. JL = MO and all angles are equal
  2. JL = MO, which can be calculated using the Law of Cosines (correct answer)
  3. Angle J = angle M = 180° - 72° - angle L
  4. The triangles have equal areas and perimeters
Explanation: When you encounter triangle congruence problems, remember that SSS (Side-Side-Side) requires proving all three corresponding sides are equal, while SAS (Side-Angle-Side) uses two sides and the included angle. You already know two pairs of corresponding sides are equal: JK = MN = 8 and KL = NO = 5. To prove congruence by SSS instead of SAS, you need the third pair: JL = MO. The Law of Cosines can calculate this missing side length using the formula c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab\cos(C). With sides 8 and 5 and the included angle of 72°, you can find JL in triangle JKL and MO in triangle MNO. Since the triangles are already congruent, these calculations will yield equal results, giving you JL = MO. Choice A is incorrect because while JL = MO is necessary, the phrase "all angles are equal" is redundant information that doesn't specifically address the SSS requirement. Choice C focuses on angle relationships, but SSS congruence doesn't involve angles at all—you need the third side measurement. Choice D mentions areas and perimeters, but these properties don't provide the specific side length needed for SSS congruence. Strategy tip: When converting between congruence postulates, identify what's missing from your target postulate. For SSS, you always need all three side lengths. The Law of Cosines is your tool for finding unknown sides when you have two sides and an included angle—exactly what SAS gives you.

Question 2

Pentagon ABCDE is similar to pentagon FGHIJ with a similarity ratio of 4:7. If the length of diagonal AC in pentagon ABCDE is 12 units, what is the length of the corresponding diagonal FH in pentagon FGHIJ?

  1. 21 units (correct answer)
  2. 16 units
  3. 28 units
  4. 48 units
Explanation: When you encounter similar polygons, remember that all corresponding linear measurements (sides, diagonals, perimeters, heights) are related by the same similarity ratio. This includes diagonals that aren't immediately obvious. Given that pentagon ABCDE is similar to pentagon FGHIJ with a ratio of 4:7, this means every linear measurement in the smaller pentagon relates to the corresponding measurement in the larger pentagon as 4:7. Since diagonal AC = 12 units in the smaller pentagon, you can set up a proportion: 47=12FH\frac{4}{7} = \frac{12}{\text{FH}} Cross-multiplying: 4×FH=7×12=844 \times \text{FH} = 7 \times 12 = 84 Therefore: FH=844=21\text{FH} = \frac{84}{4} = 21 units. A) 21 units is correct—this properly applies the similarity ratio to find the corresponding diagonal length. B) 16 units represents a common error where students add 4 to the original length (12 + 4), misunderstanding how ratios work in similar figures. C) 28 units comes from incorrectly multiplying the original diagonal by the larger ratio number: 12 × 7/3 ≈ 28. This shows confusion about which direction the ratio goes. D) 48 units results from simply multiplying 12 × 4, completely misapplying the ratio concept and ignoring the proportional relationship. Strategy tip: Always set up your similarity ratio as a fraction, then create a proportion with the known and unknown measurements. Double-check that your answer makes sense—since 7 > 4, the larger pentagon should have longer measurements than the smaller one.

Question 3

Rectangle PQRS is congruent to rectangle TUVW. If rectangle PQRS has dimensions 5 by 12 and is positioned with its longer side horizontal, while rectangle TUVW is positioned with its longer side vertical, what transformation could map PQRS onto TUVW?

  1. A 90° rotation followed by a translation (correct answer)
  2. A reflection across a diagonal line only
  3. A translation followed by a 180° rotation
  4. A uniform scaling with factor 12/5
Explanation: When you encounter questions about mapping congruent rectangles with different orientations, focus on what transformations preserve size and shape while changing position or orientation. Since both rectangles are congruent with dimensions 5 by 12, but PQRS has its longer side (12) horizontal while TUVW has its longer side vertical, you need a transformation that rotates the rectangle by 90°. A 90° rotation will flip the orientation from horizontal to vertical. However, after rotation, the rectangles likely won't be in the same position, so you'll also need a translation (sliding motion) to move one rectangle to coincide with the other. Choice A correctly identifies that you need a 90° rotation followed by a translation. This combination preserves the rectangle's dimensions while changing its orientation and position as needed. Choice B is incorrect because a single reflection across a diagonal cannot change a rectangle from horizontal to vertical orientation while maintaining congruence. Reflections flip figures but don't achieve the 90° orientation change required here. Choice C won't work because a 180° rotation would keep the longer side horizontal (just upside down), not make it vertical. The translation afterward still couldn't fix this orientation mismatch. Choice D is wrong because scaling changes the size of the figure. Since the rectangles are congruent (same size), no scaling is needed or wanted - scaling would destroy the congruence. Strategy tip: For congruent figure problems, remember that you can only use rigid transformations (rotations, reflections, translations) that preserve size and shape. If orientations differ significantly, look for rotations combined with other rigid motions.

Question 4

Two congruent isosceles triangles each have a base of 10 units and base angles of 65°. If these triangles are positioned so their bases overlap completely, what is the measure of the vertex angle in each triangle?

  1. 50° (correct answer)
  2. 115°
  3. 130°
  4. 65°
Explanation: When you encounter isosceles triangle problems, remember that the key relationship is between the base angles and the vertex angle. In any triangle, all three angles must sum to 180°. In an isosceles triangle, the two base angles are always equal. Since you're told each base angle measures 65°, you can find the vertex angle by using the angle sum property. The vertex angle plus the two base angles equals 180°: Vertex angle + 65° + 65° = 180° Vertex angle + 130° = 180° Vertex angle = 50° The information about the triangles being congruent and their bases overlapping completely is extra detail that doesn't affect the angle measurements—it's just describing how the triangles are positioned. Looking at the wrong answers: B) 115° would make the total angles 115° + 65° + 65° = 245°, which exceeds 180°. C) 130° would give you 130° + 65° + 65° = 260°, also impossible for a triangle. D) 65° would mean all three angles equal 65°, totaling 195°, which again violates the triangle angle sum rule. Answer choice A) 50° is correct because 50° + 65° + 65° = 180°. Study tip: For any isosceles triangle problem, immediately write down the angle sum equation: vertex angle + 2(base angle) = 180°. This formula will quickly lead you to the answer and help you avoid the trap answers that violate basic triangle properties.

Question 5

Square ABCD has side length 8. Square EFGH is similar to square ABCD with a scale factor of 5:4. If the area of a small square inscribed in ABCD is 16 square units, what is the area of the corresponding inscribed square in EFGH?

  1. 20 square units
  2. 25 square units (correct answer)
  3. 12.8 square units
  4. 10.24 square units
Explanation: When you encounter problems involving similar figures with inscribed shapes, the key insight is understanding how scale factors affect area measurements. Since areas involve two dimensions, the scale factor must be squared when comparing areas. Given that square EFGH is similar to square ABCD with a scale factor of 5:4, this means each side of EFGH is 54\frac{5}{4} times the corresponding side of ABCD. Since ABCD has side length 8, square EFGH has side length 8×54=108 \times \frac{5}{4} = 10. The small square inscribed in ABCD has area 16 square units, so its side length is 16=4\sqrt{16} = 4. When we scale up to the similar square EFGH, the corresponding inscribed square's side length becomes 4×54=54 \times \frac{5}{4} = 5. Therefore, its area is 52=255^2 = 25 square units. Looking at the wrong answers: Choice A (20 square units) incorrectly applies the linear scale factor to the area, calculating 16×54=2016 \times \frac{5}{4} = 20. Choice C (12.8 square units) appears to use the inverse scale factor, calculating 16×45=12.816 \times \frac{4}{5} = 12.8. Choice D (10.24 square units) uses the inverse scale factor squared: 16×(45)2=10.2416 \times \left(\frac{4}{5}\right)^2 = 10.24. The correct answer is B) 25 square units. Study tip: Remember that when similar figures have a linear scale factor of kk, their areas have a scale factor of k2k^2. Always square the scale factor when working with areas of similar figures.

Question 6

Triangle ABC is similar to triangle DEF with a scale factor of 3:2. If triangle ABC has sides of length 9, 12, and 15, and triangle DEF has sides of length 6, 8, and 10, what is the ratio of the area of triangle ABC to the area of triangle DEF?

  1. 3:2
  2. 9:4 (correct answer)
  3. 6:4
  4. 15:10
Explanation: When you encounter similar triangles with area comparisons, remember that while linear dimensions scale by the given ratio, areas scale by the square of that ratio. Given triangles ABC and DEF are similar with a scale factor of 3:2, you can verify this by comparing corresponding sides: 9÷6 = 3÷2, 12÷8 = 3÷2, and 15÷10 = 3÷2. This confirms the linear scale factor is indeed 3:2. For areas of similar figures, the ratio equals the square of the linear scale factor. Since the linear ratio is 3:2, the area ratio is (3:2)2=9:4(3:2)^2 = 9:4. This happens because area involves two dimensions, so both length and width are scaled by the factor 3:2, giving us 32×32=94\frac{3}{2} \times \frac{3}{2} = \frac{9}{4}. Looking at the wrong answers: Choice A (3:2) incorrectly uses the linear scale factor instead of squaring it for area. Choice C (6:4) appears to mix up side lengths with ratios, possibly taking the first sides (9 and 6) and the last sides (12 and 8) incorrectly. Choice D (15:10) simply takes the ratio of the longest sides, which gives the linear ratio 3:2 in a different form, but still fails to account for the area relationship. Study tip: Always remember that for similar figures, if the linear scale factor is a:ba:b, then the area ratio is a2:b2a^2:b^2 and volume ratio would be a3:b3a^3:b^3. Square the ratio for area, cube it for volume.

Question 7

In the diagram, triangles PQR and STU are congruent. The transformation that maps triangle PQR onto triangle STU includes a reflection across the y-axis followed by a translation. If P is at (-3, 2), what is the location of point P after the reflection across the y-axis?

  1. (3, 2) (correct answer)
  2. (-3, -2)
  3. (3, -2)
  4. (-3, 2)
Explanation: A reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate the same. So P(-3, 2) becomes (3, 2) after reflection across the y-axis.

Question 8

In the coordinate plane shown, triangle ABC has vertices at A(2,1), B(6,1), and C(4,5). Triangle DEF is similar to triangle ABC with vertex D at (1,2). If the scale factor from ABC to DEF is 1/2, what are the coordinates of vertex E?

  1. (3, 2) (correct answer)
  2. (1, 4)
  3. (3, 4)
  4. (5, 2)
Explanation: With scale factor 1/2, triangle DEF is half the size of triangle ABC. Vector AB = (4,0). In triangle DEF, vector DE should be half of this: (2,0). Starting from D(1,2), E = (1,2) + (2,0) = (3,2).

Question 9

In the coordinate plane shown, quadrilateral ABCD with vertices A(1,1), B(4,1), C(4,3), and D(1,3) is congruent to quadrilateral EFGH. If quadrilateral EFGH is the result of rotating ABCD 90° counterclockwise about the origin, what are the coordinates of vertex F?

  1. (-1, 4) (correct answer)
  2. (-4, 1)
  3. (1, -4)
  4. (-1, -4)
Explanation: When rotating 90° counterclockwise about the origin, the transformation rule is (x,y) → (-y,x). Vertex B(4,1) becomes (-1,4). Since the quadrilaterals are congruent and F corresponds to B, the coordinates of F are (-1,4).

Question 10

In the figure, triangle PQR is congruent to triangle STU by the ASA postulate. If angle P = 45°, PQ = 7, and angle Q = 60°, which set of measurements for triangle STU would confirm this congruence?

  1. angle S = 45°, ST = 7, angle T = 60° (correct answer)
  2. angle S = 60°, ST = 7, angle T = 45°
  3. angle S = 75°, SU = 7, angle U = 60°
  4. angle T = 45°, TU = 7, angle U = 75°
Explanation: For ASA congruence, we need two angles and the included side. In triangle PQR, we have angle P = 45°, side PQ = 7 (between angles P and Q), and angle Q = 60°. For triangle STU to be congruent by ASA, the corresponding angles and included side must be equal: angle S = 45°, ST = 7, and angle T = 60°.

Question 11

Two similar triangles have a ratio of corresponding sides of 3:5. If the area of the smaller triangle is 27 square units, what is the area of the larger triangle?

  1. 45 square units
  2. 75 square units (correct answer)
  3. 135 square units
  4. 225 square units
Explanation: When you encounter similar triangles, remember that their corresponding sides are proportional, but their areas have a special relationship that's often tested on the ISEE. Since the triangles are similar with a side ratio of 3:5, you need to find the area ratio. Here's the key insight: when similar figures have a linear ratio of a:ba:b, their areas have a ratio of a2:b2a^2:b^2. This happens because area involves two dimensions. With a side ratio of 3:5, the area ratio is 32:52=9:253^2:5^2 = 9:25. This means if the smaller triangle has area 27, you can set up the proportion: 27larger area=925\frac{27}{\text{larger area}} = \frac{9}{25}. Cross-multiplying: 9×larger area=27×25=6759 \times \text{larger area} = 27 \times 25 = 675, so the larger area is 675÷9=75675 ÷ 9 = 75 square units. Looking at the wrong answers: Choice (A) 45 represents the common mistake of using the linear ratio directly—multiplying 27 by 53\frac{5}{3} instead of squaring the ratio. Choice (C) 135 comes from multiplying 27 by 5 without considering the ratio at all. Choice (D) 225 results from incorrectly calculating 27×25327 \times \frac{25}{3}, mixing up the ratio relationship. The correct answer is (B) 75 square units. Strategy tip: Always remember to square the linear ratio when finding area ratios for similar figures. Write "linear ratio → square for area ratio" in your notes—this relationship appears frequently on geometry problems and is easy to forget under time pressure.

Question 12

Rectangle ABCD has dimensions 12 by 16. Rectangle EFGH is similar to rectangle ABCD with a scale factor of 2/3. What is the ratio of the perimeter of rectangle EFGH to the perimeter of rectangle ABCD?

  1. 2/3 (correct answer)
  2. 4/9
  3. 3/2
  4. 9/4
Explanation: When you encounter similar rectangles with a given scale factor, remember that different measurements scale differently. Linear measurements (like side lengths and perimeters) scale by the scale factor itself, while areas scale by the square of the scale factor. Since rectangle EFGH is similar to rectangle ABCD with a scale factor of 23\frac{2}{3}, each dimension of EFGH is 23\frac{2}{3} times the corresponding dimension of ABCD. Rectangle ABCD has dimensions 12 by 16, so its perimeter is 2(12+16)=562(12 + 16) = 56. Rectangle EFGH has dimensions 12×23=812 \times \frac{2}{3} = 8 by 16×23=32316 \times \frac{2}{3} = \frac{32}{3}, giving it a perimeter of 2(8+323)=2(563)=11232(8 + \frac{32}{3}) = 2(\frac{56}{3}) = \frac{112}{3}. The ratio is 112/356=1123×156=23\frac{112/3}{56} = \frac{112}{3} \times \frac{1}{56} = \frac{2}{3}. Choice A (23\frac{2}{3}) is correct because perimeter scales by the same factor as linear dimensions. Choice B (49\frac{4}{9}) represents (23)2(\frac{2}{3})^2, which would be the ratio of areas, not perimeters. This is a common mistake when students confuse how different measurements scale. Choice C (32\frac{3}{2}) is the reciprocal of the scale factor, which would apply if ABCD were similar to EFGH with scale factor 23\frac{2}{3}, rather than the other way around. Choice D (94\frac{9}{4}) is (32)2(\frac{3}{2})^2, combining both the reciprocal error and the area scaling error. Remember: for similar figures, linear measurements (perimeter, side lengths) scale by the scale factor, while areas scale by the square of the scale factor.

Question 13

Circles O and P are congruent with radius 6 cm. If circle O is centered at the origin and circle P is centered at (8, 6), what is the distance between the centers of these congruent circles?

  1. 10 cm (correct answer)
  2. 14 cm
  3. 6 cm
  4. 12 cm
Explanation: When you see a question asking for the distance between two points on a coordinate plane, you're working with the distance formula. This formula comes directly from the Pythagorean theorem and calculates the straight-line distance between any two points. The distance formula is: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Here, circle O is at the origin (0, 0) and circle P is at (8, 6). Substituting these coordinates: d=(8−0)2+(6−0)2=82+62=64+36=100=10d = \sqrt{(8 - 0)^2 + (6 - 0)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 The distance between centers is 10 cm, making (A) 10 cm correct. Looking at the wrong answers: (B) 14 cm represents adding the coordinates instead of using the distance formula (8 + 6 = 14) — a common trap when students forget the proper calculation method. (C) 6 cm is simply the radius of each circle, which has nothing to do with the distance between centers. (D) 12 cm might result from incorrectly adding the radii together (6 + 6 = 12), confusing radius properties with center distance. Strategy tip: Recognize that 8-6-10 forms a multiple of the classic 3-4-5 right triangle (specifically 2 × 3-4-5). On standardized tests, distance problems often use Pythagorean triples to make calculations cleaner. When you spot familiar number patterns like 3-4-5, 5-12-13, or their multiples, you can often solve more quickly.

Question 14

Hexagon ABCDEF is similar to hexagon GHIJKL with a scale factor of 3:4. If the perimeter of hexagon ABCDEF is 45 units, what is the perimeter of hexagon GHIJKL?

  1. 60 units (correct answer)
  2. 33.75 units
  3. 180 units
  4. 80 units
Explanation: When you encounter similar polygons with a given scale factor, you're working with proportional relationships. The key insight is that linear measurements (like side lengths and perimeters) scale directly with the scale factor, while areas scale with the square of the scale factor. Given that hexagon ABCDEF is similar to hexagon GHIJKL with a scale factor of 3:4, this means that every corresponding linear measurement in GHIJKL is 43\frac{4}{3} times the corresponding measurement in ABCDEF. Since perimeter is the sum of all side lengths, it follows this same linear scaling relationship. To find the perimeter of hexagon GHIJKL, multiply the perimeter of ABCDEF by the scale factor: 45×43=1803=6045 \times \frac{4}{3} = \frac{180}{3} = 60 units. Looking at the wrong answers: Choice B (33.75 units) results from incorrectly multiplying by 34\frac{3}{4} instead of 43\frac{4}{3} - this would give you a smaller hexagon rather than the larger one described. Choice C (180 units) comes from multiplying 45 by 4 without dividing by 3, ignoring the ratio aspect of the scale factor. Choice D (80 units) doesn't follow from any clear mathematical relationship and likely represents a calculation error. The correct answer is A) 60 units. Remember this pattern: when polygons are similar with scale factor a:b, linear measurements (perimeter, side length, height) scale by the factor ba\frac{b}{a}, while areas scale by (ba)2\left(\frac{b}{a}\right)^2. Always pay attention to which polygon the scale factor describes first.

Question 15

Two similar right triangles have legs in the ratio 2:5. If the smaller triangle has legs of length 6 and 8, what is the length of the hypotenuse of the larger triangle?

  1. 12.5 units
  2. 15 units
  3. 25 units (correct answer)
  4. 30 units
Explanation: When you encounter similar triangles, remember that all corresponding sides are in the same ratio. This means if you know the ratio between any pair of corresponding sides, that same ratio applies to all other corresponding pairs. Start by finding the scale factor between the triangles. You're told the legs are in the ratio 2:5, meaning the smaller triangle's sides are 2 parts while the larger triangle's sides are 5 parts. This gives us a scale factor of 52=2.5\frac{5}{2} = 2.5 from smaller to larger. First, find the hypotenuse of the smaller triangle using the Pythagorean theorem: c2=62+82=36+64=100c^2 = 6^2 + 8^2 = 36 + 64 = 100, so c=10c = 10 units. Now apply the scale factor to find the larger triangle's hypotenuse: 10×2.5=2510 \times 2.5 = 25 units. Looking at the wrong answers: (A) 12.5 represents half of the correct answer, possibly from using the wrong direction for the ratio. (B) 15 might come from incorrectly adding 5 to the smaller hypotenuse instead of using proportional scaling. (D) 30 could result from multiplying the hypotenuse by 3 instead of 2.5, perhaps confusing the scale factor. The correct answer is (C) 25 units. Study tip: With similar triangles, always establish the scale factor first by comparing corresponding sides, then apply that same factor to find any unknown measurement. Don't forget to use the Pythagorean theorem when you need to find a missing side before scaling.

Question 16

Rhombus JKLM is congruent to rhombus NOPQ. If the diagonals of rhombus JKLM are 16 and 12 units, and the diagonals intersect at right angles, what is the area of rhombus NOPQ?

  1. 96 square units (correct answer)
  2. 192 square units
  3. 48 square units
  4. 144 square units
Explanation: When you encounter congruent polygons, remember that they have identical measurements in all corresponding parts. This means congruent rhombuses have the same area, so finding the area of one gives you the area of both. For any rhombus, the area formula is: Area=12×d1×d2\text{Area} = \frac{1}{2} \times d_1 \times d_2, where d1d_1 and d2d_2 are the lengths of the diagonals. The fact that diagonals intersect at right angles is actually a defining property of all rhombuses, so this information confirms you're working with a valid rhombus. Using the given diagonal lengths of 16 and 12 units for rhombus JKLM: Area=12×16×12=12×192=96\text{Area} = \frac{1}{2} \times 16 \times 12 = \frac{1}{2} \times 192 = 96 square units Since the rhombuses are congruent, rhombus NOPQ has the same area: 96 square units. Looking at the wrong answers: Choice B (192) represents the common error of forgetting to multiply by 12\frac{1}{2} – this would be 16×1216 \times 12 without applying the correct formula. Choice C (48) suggests incorrectly using 14\frac{1}{4} instead of 12\frac{1}{2} in the formula. Choice D (144) might result from mistakenly using 34\frac{3}{4} or some other calculation error. The correct answer is A. Study tip: Always remember that rhombus area uses 12×d1×d2\frac{1}{2} \times d_1 \times d_2, and congruent figures have identical areas. Don't let the mention of perpendicular diagonals throw you – that's just confirming the shape's properties, not adding complexity to your calculation.

Question 17

Trapezoid ABCD is similar to trapezoid EFGH with parallel sides AB∥CD and EF∥GH. If AB = 8, CD = 12, EF = 6, and the height of trapezoid ABCD is 5, what is the height of trapezoid EFGH?

  1. 3.75 units (correct answer)
  2. 4.5 units
  3. 6.25 units
  4. 7.5 units
Explanation: When you encounter similar trapezoids, remember that all corresponding linear measurements are proportional by the same scale factor. This includes the parallel sides, legs, and height. To find the scale factor between these trapezoids, compare corresponding parallel sides. Since trapezoid ABCD is similar to trapezoid EFGH, we can use the parallel sides AB and EF: EFAB=68=34\frac{EF}{AB} = \frac{6}{8} = \frac{3}{4} We can verify this scale factor using the other pair of parallel sides. If CD corresponds to GH, then: GH=CD×34=12×34=9GH = CD \times \frac{3}{4} = 12 \times \frac{3}{4} = 9 Since the height also scales by the same factor: Height of EFGH=5×34=3.75 units\text{Height of EFGH} = 5 \times \frac{3}{4} = 3.75 \text{ units} Looking at the wrong answers: Choice B (4.5) would result from incorrectly using EF+GHAB+CD=6+98+12=1520=34\frac{EF + GH}{AB + CD} = \frac{6 + 9}{8 + 12} = \frac{15}{20} = \frac{3}{4} and then multiplying by 6 instead of 5. Choice C (6.25) comes from using the reciprocal scale factor 43\frac{4}{3} and calculating 5×43−5125 \times \frac{4}{3} - \frac{5}{12}. Choice D (7.5) results from simply using 32\frac{3}{2} as a scale factor. The correct answer is A (3.75 units). Study tip: Always identify the scale factor first by comparing any pair of corresponding sides, then apply that same factor to all linear measurements including height, perimeter parts, and diagonal lengths.

Question 18

Given two triangles with sides labeled, which statement about their congruence is correct?

  1. Congruent by SSS: 5,7,85,7,8 match 5,7,85,7,8 (correct answer)
  2. Similar only: sides are proportional 5:7:8=10:14:165:7:8=10:14:16
  3. Congruent by dilation factor 22
  4. Neither: one angle differs so lengths cannot match
Explanation: This question tests upper-level ISEE skills in reasoning about congruence and similarity. Congruence involves identical figures where all sides and angles are equal, while similarity involves proportional sides and equal angles. The problem presents two triangles, and based on the correct answer A, they have identical side lengths of 5, 7, and 8 units, which proves congruence by the SSS (Side-Side-Side) criterion. The correct choice A identifies that when all three corresponding sides are equal, the triangles are congruent. A common misconception would be choosing option B, thinking the triangles are only similar when they are actually congruent, or option C, misunderstanding that dilation creates similarity, not congruence. To help students: Emphasize the difference between congruent (same size and shape) and similar (same shape, different size), practice applying congruence criteria (SSS, SAS, ASA), and use physical triangles to demonstrate that equal sides guarantee congruence.

Question 19

Right triangle ABC is congruent to right triangle DEF. If the right angle in triangle ABC is at vertex C, and AC = 5, BC = 12, what is the length of the hypotenuse in triangle DEF?

  1. 13 units (correct answer)
  2. 17 units
  3. 15 units
  4. 12 units
Explanation: When you encounter congruent triangles, remember that corresponding sides and angles are equal. This means if you can find a measurement in one triangle, that same measurement exists in its congruent partner. Since triangle ABC has a right angle at C, you need to find the hypotenuse AB using the given legs AC = 5 and BC = 12. Apply the Pythagorean theorem: a2+b2=c2a^2 + b^2 = c^2, where c is the hypotenuse. Substituting the values: 52+122=c25^2 + 12^2 = c^2, which gives us 25+144=16925 + 144 = 169, so c=169=13c = \sqrt{169} = 13. Since triangles ABC and DEF are congruent, triangle DEF must also have a hypotenuse of 13 units, making A correct. Looking at the wrong answers: B (17 units) might tempt you if you incorrectly added the legs instead of using the Pythagorean theorem, or made an arithmetic error in your calculations. C (15 units) could result from miscalculating 52+1225^2 + 12^2 or confusing this with a different Pythagorean triple. D (12 units) is simply one of the given leg lengths, which might catch you if you confused which side is the hypotenuse or misunderstood what the question was asking for. Study tip: Memorize common Pythagorean triples like 3-4-5, 5-12-13, and 8-15-17. Recognizing that 5 and 12 are part of the 5-12-13 triple would let you identify the hypotenuse immediately without calculation, saving valuable time on the exam.

Question 20

How can you prove these shapes are congruent through transformations on the tiled hallway design?

  1. Dilate one rhombus by factor cfrac{3}{2}
  2. Translate, then rotate to overlap exactly (correct answer)
  3. Rotate, then dilate to match size
  4. Reflect, then dilate to match angles
Explanation: This question tests upper-level ISEE skills in reasoning about congruence and similarity. Congruence involves identical figures where all sides and angles are equal, while similarity involves proportional sides and equal angles. The question asks how to prove shapes are congruent through transformations, focusing on rigid motions that preserve size and shape. The correct choice B suggests using translation followed by rotation to make the shapes overlap exactly, which are both rigid transformations that maintain congruence. A common misconception would be choosing options that include dilation (A, C, or D), which changes size and thus cannot prove congruence. To help students: Emphasize that congruence proofs require only rigid transformations (translation, rotation, reflection), practice sequencing transformations to map one figure onto another, and use tracing paper or digital tools to visualize how rigid motions preserve congruence.