Apply Congruency And Similarity - Praxis Core Math

1. In $\triangle LMN$, $m\angle L = 40^\circ$ and $m\angle M = 60^\circ$. In $\triangle PQR$, $m\angle P = 40^\circ$ and $m\angle R = 80^\circ$. Which statement is true about the two triangles?
   Answer: They are similar. First, find the third angle in each triangle. The sum of angles in a triangle is 180 degrees. In $\triangle LMN$, $m\angle N = 180^\circ - 40^\circ - 60^\circ = 80^\circ$. In $\triangle PQR$, $m\angle Q = 180^\circ - 40^\circ - 80^\circ = 60^\circ$. So, $\triangle LMN$ has angles 40, 60, 80 and $\triangle PQR$ has angles 40, 60, 80. Since all three pairs of corresponding angles are congruent, the triangles are similar by the Angle-Angle-Angle (AAA) or Angle-Angle (AA) similarity criterion. We cannot determine if they are congruent as no side lengths are given.
2. The side lengths of $\triangle PQR$ are 5, 12, and 13. A second triangle, $\triangle STU$, is similar to $\triangle PQR$, and its longest side is 39. What is the perimeter of $\triangle STU$?
   Answer: 90 First, find the perimeter of $\triangle PQR$: $5 + 12 + 13 = 30$. The longest side of $\triangle PQR$ is 13. The corresponding longest side of $\triangle STU$ is 39. The scale factor from $\triangle PQR$ to $\triangle STU$ is the ratio of these corresponding sides: $k = \frac{39}{13} = 3$. The ratio of the perimeters of two similar triangles is equal to the scale factor of their side lengths. Therefore, the perimeter of $\triangle STU$ is the perimeter of $\triangle PQR$ multiplied by the scale factor: $30 \times 3 = 90$.
3. A polygon is transformed on a coordinate plane. Which of the following transformations does NOT necessarily result in an image that is congruent to the original polygon?
   Answer: A dilation with a scale factor of 2. Congruent figures have the same size and shape. Reflections, translations, and rotations are rigid transformations (isometries), which means they preserve side lengths and angle measures. Therefore, the image after these transformations is always congruent to the original figure. A dilation, however, changes the size of the figure by a given scale factor. Unless the scale factor is 1 or -1, the resulting image will be similar but not congruent to the original polygon.
4. In $\triangle XYZ$, a line segment PQ is drawn such that P is on side XY and Q is on side XZ. If PQ is parallel to YZ, which statement guarantees that $\triangle XPQ$ is similar to $\triangle XYZ$?
   Answer: No additional information is needed; the triangles are always similar. If a line segment (PQ) is drawn parallel to one side (YZ) of a triangle (XYZ) intersecting the other two sides, it creates a smaller triangle (XPQ) that is similar to the original triangle. This is because the parallel lines create corresponding angles that are congruent: $\angle XPQ \cong \angle XYZ$ and $\angle XQP \cong \angle XZY$. Since $\angle PXQ$ is the same as $\angle YXZ$ (reflexive property), the triangles are similar by the Angle-Angle (AA) similarity postulate. No further conditions are required.
5. Triangle ABC is congruent to triangle DEF. The perimeter of triangle ABC is 32. If DE = 10 and EF = 12, what is the length of side AC?
   Answer: 10 Since $\triangle ABC \cong \triangle DEF$, their corresponding sides are equal in length. This means $AB = DE$, $BC = EF$, and $AC = DF$. We are given DE = 10 and EF = 12, so AB = 10 and BC = 12. The perimeter of triangle ABC is given as 32. The perimeter is the sum of the side lengths: $AB + BC + AC = 32$. Substituting the known values: $10 + 12 + AC = 32$. This simplifies to $22 + AC = 32$. Subtracting 22 from both sides gives $AC = 10$.
6. A rectangular garden has a length of 25 feet and a width of 15 feet. A landscape designer creates a scale drawing of the garden where the width on the drawing is 6 inches. What is the length of the garden on the scale drawing?
   Answer: 10 inches The actual garden and the scale drawing are similar rectangles. The ratio of the width to the length in the actual garden must be equal to the ratio of the width to the length in the drawing. Let $L$ be the length on the drawing. Set up a proportion: $\frac{\text{drawing width}}{\text{actual width}} = \frac{\text{drawing length}}{\text{actual length}}$. Or, more simply, $\frac{\text{drawing length}}{\text{drawing width}} = \frac{\text{actual length}}{\text{actual width}}$. So, $\frac{L}{6} = \frac{25}{15}$. Simplifying the fraction gives $\frac{L}{6} = \frac{5}{3}$. Multiply both sides by 6 to solve for $L$: $L = \frac{5}{3} \times 6 = 10$. The length on the drawing is 10 inches.
7. Two similar pyramids have heights of 10 cm and 15 cm. If the volume of the smaller pyramid is 400 cm$^3$, what is the volume of the larger pyramid?
   Answer: 1350 cm$^3$ The scale factor of the linear dimensions (like height) from the smaller pyramid to the larger is $k = \frac{15}{10} = \frac{3}{2}$. The ratio of the volumes of two similar solids is the cube of the scale factor, $k^3$. So, the ratio of the volumes is $(\frac{3}{2})^3 = \frac{27}{8}$. To find the volume of the larger pyramid, multiply the volume of the smaller pyramid by this ratio: Volume(large) = $\frac{27}{8} \times 400 = 27 \times 50 = 1350$ cm$^3$.
8. Rectangle A has a length of 12 cm and a width of 8 cm. Rectangle B is similar to Rectangle A and has a length of 18 cm. What is the perimeter of Rectangle B?
   Answer: 60 cm First, find the scale factor from Rectangle A to Rectangle B using the corresponding lengths: $k = \frac{18}{12} = \frac{3}{2}$. The width of Rectangle B will be the width of Rectangle A multiplied by the scale factor: Width B = $8 \times \frac{3}{2} = 12$ cm. Now, calculate the perimeter of Rectangle B: Perimeter = $2 \times (\text{length} + \text{width}) = 2 \times (18 + 12) = 2 \times 30 = 60$ cm. Alternatively, find the perimeter of Rectangle A: $2 \times (12 + 8) = 40$ cm. Then multiply by the scale factor: $40 \times \frac{3}{2} = 60$ cm.
9. In the figure shown, quadrilateral $ABCD$ is similar to quadrilateral $EFGH$. Given $AB = 10$, $BC = 14$, $EF = 15$, and the perimeter of $ABCD$ is $48$, what is the perimeter of $EFGH$?
   Answer: $72$ The linear scale factor from $ABCD$ to $EFGH$ is $EF/AB = 15/10 = 3/2$. Perimeters scale by the same factor: $48 \cdot 3/2 = 72$. B (32) uses $48 \cdot 2/3$ (inverted ratio). C (108) uses squared ratio ($48 \cdot 9/4$). D (64) uses $48 + 16$.
10. In the figure, two similar triangles have areas of $48$ and $108$ square units. If the shortest side of the smaller triangle is $8$ units, what is the length of the corresponding shortest side of the larger triangle?
    Answer: $12$ For similar figures, area ratio = (linear ratio)². So $(k)^2 = 108/48 = 9/4$, giving $k = 3/2$. The corresponding side is $8 \cdot 3/2 = 12$. B (18) uses area ratio directly: $8 \cdot 108/48$. C (15) uses $8 + 7$ or linear difference. D (10.4) uses $\sqrt{108/48} \cdot 8$ incorrectly computed.
11. In parallelogram QRST, a diagonal QS is drawn. Which of the following statements must be true regarding the two triangles formed, $\triangle QRS$ and $\triangle STQ$?
    Answer: The triangles are congruent by the Side-Side-Side (SSS) postulate. In a parallelogram, opposite sides are equal in length. Therefore, $QR = ST$ and $RS = TQ$. The diagonal $QS$ is a side shared by both triangles, so $QS = SQ$ by the reflexive property. Since all three pairs of corresponding sides are equal, the triangles $\triangle QRS$ and $\triangle STQ$ are congruent by the Side-Side-Side (SSS) postulate.
12. Triangle ABC is similar to triangle DEF. The length of side AB is 8 inches and the length of the corresponding side DE is 12 inches. If the area of triangle ABC is 40 square inches, what is the area of triangle DEF?
    Answer: 90 square inches The ratio of corresponding side lengths (the scale factor) from triangle ABC to triangle DEF is $k = \frac{12}{8} = \frac{3}{2}$. The ratio of the areas of two similar figures is the square of the scale factor, $k^2$. Therefore, the ratio of the areas is $(\frac{3}{2})^2 = \frac{9}{4}$. To find the area of triangle DEF, multiply the area of triangle ABC by this ratio: Area(DEF) = Area(ABC) \times \frac{9}{4} = 40 \times \frac{9}{4} = 10 \times 9 = 90\) square inches.
13. Triangle ABC has vertices with coordinates A(0,0), B(4,0), and C(0,3). Triangle DEF is formed by applying a dilation centered at the origin with a scale factor of 2.5 to triangle ABC. What is the length of side EF?
    Answer: 12.5 First, find the length of the corresponding side BC in triangle ABC. Since the vertices are on the axes, triangle ABC is a right triangle. We can use the Pythagorean theorem or the distance formula. The length of AB is 4 and the length of AC is 3. The length of the hypotenuse BC is $\sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$. A dilation multiplies all lengths by the scale factor. The side EF in triangle DEF corresponds to the side BC in triangle ABC. Therefore, the length of EF is the length of BC multiplied by the scale factor: $5 \times 2.5 = 12.5$.
14. Use the figure shown. In $\triangle ABC$, point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ such that $\overline{DE} \parallel \overline{BC}$. If $AD = x$, $DB = x + 2$, $AE = 3$, and $EC = 5$, what is the value of $x$?
    Answer: $3$ By the Triangle Proportionality Theorem, $AD/DB = AE/EC$, so $x/(x+2) = 3/5$. Cross-multiplying: $5x = 3x + 6$, so $2x = 6$, $x = 3$. B (5) uses $EC$ directly. C (2) from miscomputing. D ($10/3$) comes from solving $x/(x+2) = 5/3$ (inverted ratio).
15. Two solid spheres are mathematically similar. The ratio of their surface areas is 4:9. If the volume of the smaller sphere is 32$\pi$ cubic units, what is the volume of the larger sphere?
    Answer: 108$\pi$ cubic units Let the scale factor of the radii be $k$. The ratio of the surface areas is $k^2$ and the ratio of the volumes is $k^3$. Given the area ratio is $\frac{4}{9}$, we have $k^2 = \frac{4}{9}$, so $k = \sqrt{\frac{4}{9}} = \frac{2}{3}$. This is the ratio from larger to smaller, so let's use the ratio from smaller to larger, which is $\frac{3}{2}$. The ratio of the volumes is $k^3 = (\frac{3}{2})^3 = \frac{27}{8}$. Let $V_L$ be the volume of the larger sphere and $V_S$ be the volume of the smaller sphere. $\frac{V_L}{V_S} = \frac{27}{8}$. So, $V_L = \frac{27}{8} \times V_S = \frac{27}{8} \times 32\pi = 27 \times 4\pi = 108\pi$ cubic units.
16. Use the figure shown. In right triangle $ABC$, $\angle C = 90°$. Altitude $\overline{CD}$ is drawn to hypotenuse $\overline{AB}$. If $AC = 12$ and $AB = 18$, what is the length of $\overline{AD}$?
    Answer: $8$ By the geometric mean (leg) theorem, $AC^2 = AD \cdot AB$. So $12^2 = AD \cdot 18$, giving $AD = 144/18 = 8$. A (6) is $AB - AC$. C (9) is half of $AB$ (treating $D$ as midpoint). D comes from a misapplication of ratios.
17. Two similar triangles have a scale factor of 1:3. If the area of the smaller triangle is 7 square units, what is the area of the larger triangle?
    Answer: 63 square units When two triangles are similar with a scale factor of 1:3, the ratio of their areas is the square of the scale factor. The area ratio is $1^2 : 3^2 = 1:9$. This means the larger triangle has an area 9 times that of the smaller triangle. Area of larger triangle = $9 \times 7 = 63$ square units.
18. The areas of two similar polygons are 50 square inches and 98 square inches. If the perimeter of the smaller polygon is 30 inches, what is the perimeter of the larger polygon?
    Answer: 42 inches First, find the ratio of the areas: $\frac{98}{50} = \frac{49}{25}$. This ratio is equal to the square of the scale factor of the side lengths ($k^2$). So, $k^2 = \frac{49}{25}$. The scale factor $k$ is the square root of this ratio: $k = \sqrt{\frac{49}{25}} = \frac{7}{5}$. The ratio of the perimeters is equal to the scale factor $k$. To find the perimeter of the larger polygon, multiply the perimeter of the smaller polygon by the scale factor: Perimeter(large) = $30 \times \frac{7}{5} = 6 \times 7 = 42$ inches.
19. Two triangles are similar. The sides of the first triangle are 7, 8, and 10. The perimeter of the second triangle is 100. What is the length of the shortest side of the second triangle?
    Answer: 28 First, find the perimeter of the first triangle: $7 + 8 + 10 = 25$. The ratio of the perimeters is equal to the scale factor of the corresponding sides. The scale factor from the first triangle to the second is $k = \frac{\text{Perimeter 2}}{\text{Perimeter 1}} = \frac{100}{25} = 4$. To find the length of any side of the second triangle, multiply the corresponding side of the first triangle by the scale factor. The shortest side of the first triangle is 7. Therefore, the shortest side of the second triangle is $7 \times 4 = 28$.
20. On a sunny day, a 6-foot tall person casts an 8-foot long shadow. At the same time, a nearby tree casts a 60-foot long shadow. How tall is the tree?
    Answer: 45 feet The person and their shadow, and the tree and its shadow, form two similar right triangles. The ratio of height to shadow length is the same for both. Let $h$ be the height of the tree. Set up the proportion: $\frac{\text{person's height}}{\text{person's shadow}} = \frac{\text{tree's height}}{\text{tree's shadow}}$. So, $\frac{6}{8} = \frac{h}{60}$. Simplify the fraction: $\frac{3}{4} = \frac{h}{60}$. To solve for $h$, multiply both sides by 60: $h = \frac{3}{4} \times 60 = 3 \times 15 = 45$. The tree is 45 feet tall.

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