Similarity of Triangles

{"questions":[{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 12$, $BC = 8$, and $PM = 9$, what is $MK$?[[image-1]][[choice-6]]","widgets":{"image-1":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-6":{"type":"choice","options":["$12$","$6$","$4$","$16$"],"explanations":["","",",",""],"answer":[1]}},"hints":["Corresponding sides of similar triangles are proportional: $\\dfrac{PM}{AB} = \\dfrac{MK}{BC}$.","$\\dfrac{PM}{AB} = \\dfrac{9}{12} = \\dfrac{3}{4}$, so $MK = BC \\cdot \\dfrac{3}{4} = 8 \\cdot \\dfrac{3}{4} = 6$."]},{"content":"Two triangles are similar with scale factor $k = 4$ from the smaller to the larger triangle. The smaller triangle has perimeter $18$. What is the perimeter of the larger triangle?[[choice-35]]","widgets":{"choice-35":{"type":"choice","options":["$36$","$4,5$","$72$","$9$"],"answer":[2]}},"hints":["The perimeters of similar triangles have the same ratio as their corresponding sides.","If the scale factor is $k = 4$, then $P_2 = P_1 \\cdot k = 18 \\cdot 4 = 72$."]},{"content":"Two triangles are similar with scale factor $k = 3$ from the smaller to the larger triangle. The smaller triangle has area $5\\ \\text{cm}^2$. What is the area of the larger triangle?[[choice-74]]","widgets":{"choice-74":{"type":"choice","options":["$10\\ \\text{cm}^2$","$15\\ \\text{cm}^2$","$25\\ \\text{cm}^2$","$45\\ \\text{cm}^2$"],"answer":[3]}},"hints":["The areas of similar figures are in the ratio $k^2$.","$S_2 = S_1 \\cdot k^2 = 5 \\cdot 3^2 = 5 \\cdot 9 = 45\\ \\text{cm}^2$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. An altitude of the smaller triangle is $h_1 = 15$, and the scale factor from smaller to larger is $k = \\dfrac{5}{3}$. What is the corresponding altitude $h$ of the larger triangle?[[image-114]][[choice-130]]","widgets":{"image-114":{"type":"image","url":""},"choice-130":{"type":"choice","options":["$25$","$9$","$10$","$15$"],"answer":[0]}},"hints":["All lengths in similar triangles, including sides, altitudes, and medians, are multiplied by the scale factor $k$.","$h = 15 \\cdot \\dfrac{5}{3} = 25$."]},{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 18$, $BC = 10$, and $PM = 12$, what is $MK$?[[image-2]][[choice-7]]","widgets":{"image-2":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-7":{"type":"choice","options":["$\\frac{20}{3}$","$\\frac{25}{3}$","$\\frac{40}{3}$","$\\frac{15}{2}$"],"answer":[1]}},"hints":["Use the proportion between corresponding sides: $\\dfrac{PM}{AB} = \\dfrac{MK}{BC}$.","$\\dfrac{PM}{AB}=\\dfrac{12}{18}=\\dfrac{2}{3}$, so $MK=10\\cdot\\dfrac{2}{3}=\\dfrac{20}{3}$."]},{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 14$, $BC = 21$, and $PM = 8$, what is $MK$?[[image-3]][[choice-8]]","widgets":{"image-3":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-8":{"type":"choice","options":["$12$","$\\frac{3}{2}$","$\\frac{2}{3}$","$\\frac{7}{2}$"],"answer":[0]}},"hints":["Find the scale factor using a pair of corresponding sides: $\\dfrac{PM}{AB}$.","$\\dfrac{PM}{AB}=\\dfrac{8}{14}=\\dfrac{4}{7}$, so $MK=BC\\cdot\\dfrac{4}{7}=21\\cdot\\dfrac{4}{7}=12$."]},{"content":"Two triangles are similar with scale factor $k = 2$ from the smaller to the larger triangle. The smaller triangle has perimeter $23$. What is the perimeter of the larger triangle?[[choice-36]]","widgets":{"choice-36":{"type":"choice","options":["$46$","$11,5$","$25$","$92$"],"answer":[0]}},"hints":["The perimeter is multiplied by $k$, just like the side lengths.","$P_2 = P_1\\cdot k = 23\\cdot 2 = 46$."]},{"content":"Two triangles are similar with scale factor $k = \\dfrac{3}{2}$ from the smaller to the larger triangle. The smaller triangle has perimeter $28$. What is the perimeter of the larger triangle?[[choice-37]]","widgets":{"choice-37":{"type":"choice","options":["$42$","$18\\frac{2}{3}$","$56$","$21$"],"answer":[0]}},"hints":["The ratio of the perimeters of similar triangles equals the scale factor $k$.","$P_2 = 28\\cdot\\dfrac{3}{2} = 42$."]},{"content":"Two triangles are similar with scale factor $k = 2$ from the smaller to the larger triangle. The smaller triangle has area $12\\ \\text{cm}^2$. What is the area of the larger triangle?[[choice-75]]","widgets":{"choice-75":{"type":"choice","options":["$24\\ \\text{cm}^2$","$36\\ \\text{cm}^2$","$48\\ \\text{cm}^2$","$72\\ \\text{cm}^2$"],"answer":[2]}},"hints":["The areas of similar figures are in the ratio $k^2$.","$S_2 = 12\\cdot 2^2 = 12\\cdot 4 = 48\\ \\text{cm}^2$."]},{"content":"Two triangles are similar with scale factor $k = \\dfrac{1}{2}$ from the smaller to the larger triangle. The larger triangle has area $32\\ \\text{cm}^2$. What is the area of the smaller triangle?[[choice-76]]","widgets":{"choice-76":{"type":"choice","options":["$16\\ \\text{cm}^2$","$8\\ \\text{cm}^2$","$4\\ \\text{cm}^2$","$2\\ \\text{cm}^2$"],"answer":[1]}},"hints":["If the length scale factor is $k$, the areas are in the ratio $k^2$.","$S_{\\text{smaller}} = 32\\cdot\\left(\\dfrac{1}{2}\\right)^2 = 32\\cdot\\dfrac{1}{4}=8\\ \\text{cm}^2$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. A median of the smaller triangle is $m_1 = 12$, and the scale factor from smaller to larger is $k = \\dfrac{7}{4}$. What is the corresponding median $m$ of the larger triangle?[[image-115]][[choice-131]]","widgets":{"image-115":{"type":"image","url":""},"choice-131":{"type":"choice","options":["$21$","$\\frac{48}{7}$","$16$","$\\frac{28}{3}$"],"answer":[0]}},"hints":["Medians are multiplied by the same scale factor as corresponding sides: $k$.","$m = 12\\cdot\\dfrac{7}{4} = 21$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. An angle bisector segment in the smaller triangle has length $l_1 = 9$, and the scale factor from smaller to larger is $k = \\dfrac{4}{3}$. What is the corresponding length $l$ in the larger triangle?[[image-116]][[choice-132]]","widgets":{"image-116":{"type":"image","url":""},"choice-132":{"type":"choice","options":["$12$","$6,75$","$10$","$15$"],"answer":[0]}},"hints":["All corresponding lengths, including angle bisector segments, are multiplied by $k$.","$l = 9\\cdot\\dfrac{4}{3} = 12$."]}]}

{"questions":[{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 15$, $BC = 12$, and $PM = 10$, what is $MK$?[[image-1]][[choice-6]]","widgets":{"image-1":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-6":{"type":"choice","options":["$6$","$8$","$9$","$18$"],"answer":[1]}},"step":1,"hints":["The ratios of corresponding sides in similar triangles are equal: $\\dfrac{PM}{AB} = \\dfrac{MK}{BC}$.","$\\dfrac{PM}{AB} = \\dfrac{10}{15} = \\dfrac{2}{3}$, so $MK = 12 \\cdot \\dfrac{2}{3} = 8$."]},{"content":"Two triangles are similar with scale factor $k = 5$ from the smaller to the larger triangle. The smaller triangle has perimeter $16$. What is the perimeter of the larger triangle?[[choice-35]]","widgets":{"choice-35":{"type":"choice","options":["$80$","$21$","$3,2$","$64$"],"answer":[0]}},"step":1,"hints":["The perimeters of similar triangles have the same ratio as their corresponding sides.","If $k=5$, then $P_2 = P_1 \\cdot k = 16 \\cdot 5 = 80$."]},{"content":"Two triangles are similar with scale factor $k = 2$ from the smaller to the larger triangle. The smaller triangle has area $7\\ \\text{cm}^2$. What is the area of the larger triangle?[[choice-74]]","widgets":{"choice-74":{"type":"choice","options":["$14\\ \\text{cm}^2$","$21\\ \\text{cm}^2$","$28\\ \\text{cm}^2$","$35\\ \\text{cm}^2$"],"answer":[2]}},"step":1,"hints":["The areas of similar figures are in the ratio $k^2$.","$S_2 = 7 \\cdot 2^2 = 7 \\cdot 4 = 28\\ \\text{cm}^2$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. An altitude of the smaller triangle is $h_1 = 12$, and the scale factor from smaller to larger is $k = \\dfrac{7}{3}$. Find the corresponding altitude $h$ of the larger triangle.[[image-114]][[choice-130]]","widgets":{"image-114":{"type":"image","url":""},"choice-130":{"type":"choice","options":["$20$","$24$","$28$","$36$"],"answer":[2]}},"step":1,"hints":["All corresponding lengths in similar triangles, including altitudes, medians, and sides, are multiplied by $k$.","$h = 12 \\cdot \\dfrac{7}{3} = 28$."]},{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 20$, $BC = 18$, and $PM = 12$, what is $MK$?[[image-2]][[choice-7]]","widgets":{"image-2":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-7":{"type":"choice","options":["$\\dfrac{27}{5}$","$\\dfrac{54}{5}$","$\\dfrac{45}{4}$","$\\dfrac{36}{5}$"],"answer":[1]}},"step":1,"hints":["Use the proportion between corresponding sides: $\\dfrac{PM}{AB} = \\dfrac{MK}{BC}$.","$\\dfrac{PM}{AB}=\\dfrac{12}{20}=\\dfrac{3}{5}$, so $MK=18\\cdot\\dfrac{3}{5}=\\dfrac{54}{5}$."]},{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 16$, $BC = 24$, and $PM = 10$, what is $MK$?[[image-3]][[choice-8]]","widgets":{"image-3":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-8":{"type":"choice","options":["$12$","$15$","$\\dfrac{25}{3}$","$\\dfrac{40}{3}$"],"answer":[1]}},"step":1,"hints":["Find the scale factor using corresponding sides: $\\dfrac{PM}{AB}$.","$\\dfrac{PM}{AB}=\\dfrac{10}{16}=\\dfrac{5}{8}$, so $MK=24\\cdot\\dfrac{5}{8}=15$."]},{"content":"Two triangles are similar with scale factor $k = 3$ from the smaller to the larger triangle. The smaller triangle has perimeter $19$. What is the perimeter of the larger triangle?[[choice-36]]","widgets":{"choice-36":{"type":"choice","options":["$57$","$38$","$63$","$6,33$"],"answer":[0]}},"step":1,"hints":["The perimeter is multiplied by $k$, just like the side lengths.","$P_2 = P_1\\cdot k = 19\\cdot 3 = 57$."]},{"content":"Two triangles are similar with scale factor $k = \\dfrac{5}{2}$ from the smaller to the larger triangle. The smaller triangle has perimeter $24$. What is the perimeter of the larger triangle?[[choice-37]]","widgets":{"choice-37":{"type":"choice","options":["$60$","$48$","$30$","$9,6$"],"answer":[0]}},"step":1,"hints":["The ratio of the perimeters of similar triangles equals the scale factor $k$.","$P_2 = 24\\cdot\\dfrac{5}{2} = 60$."]},{"content":"Two triangles are similar with scale factor $k = 3$ from the smaller to the larger triangle. The smaller triangle has area $6\\ \\text{cm}^2$. What is the area of the larger triangle?[[choice-75]]","widgets":{"choice-75":{"type":"choice","options":["$18\\ \\text{cm}^2$","$36\\ \\text{cm}^2$","$54\\ \\text{cm}^2$","$72\\ \\text{cm}^2$"],"answer":[2]}},"step":1,"hints":["The areas of similar figures are in the ratio $k^2$.","$S_2 = 6\\cdot 3^2 = 6\\cdot 9 = 54\\ \\text{cm}^2$."]},{"content":"Two triangles are similar with scale factor $k = \\dfrac{2}{3}$ from the smaller to the larger triangle. The larger triangle has area $45\\ \\text{cm}^2$. What is the area of the smaller triangle?[[choice-76]]","widgets":{"choice-76":{"type":"choice","options":["$20\\ \\text{cm}^2$","$15\\ \\text{cm}^2$","$30\\ \\text{cm}^2$","$10\\ \\text{cm}^2$"],"answer":[1]}},"step":1,"hints":["If the length scale factor is $k$, the areas are in the ratio $k^2$.","$S_{\\text{smaller}} = 45\\cdot\\left(\\dfrac{2}{3}\\right)^2 = 45\\cdot\\dfrac{4}{9}=20\\ \\text{cm}^2$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. A median of the smaller triangle is $m_1 = 14$, and the scale factor from smaller to larger is $k = \\dfrac{9}{7}$. Find the corresponding median $m$ of the larger triangle.[[image-115]][[choice-131]]","widgets":{"image-115":{"type":"image","url":""},"choice-131":{"type":"choice","options":["$18$","$16$","$\\dfrac{98}{9}$","$21$"],"answer":[0]}},"step":1,"hints":["Medians are multiplied by the same scale factor as corresponding sides: $k$.","$m = 14\\cdot\\dfrac{9}{7} = 18$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. An angle bisector segment in the smaller triangle has length $l_1 = 12$, and the scale factor from smaller to larger is $k = \\dfrac{11}{6}$. Find the corresponding length $l$ in the larger triangle.[[image-116]][[choice-132]]","widgets":{"image-116":{"type":"image","url":""},"choice-132":{"type":"choice","options":["$22$","$18$","$16$","$\\dfrac{66}{11}$"],"answer":[0]}},"step":1,"hints":["All corresponding lengths, including angle bisector segments, are multiplied by $k$.","$l = 12\\cdot\\dfrac{11}{6} = 22$."]}]}

{"questions":[{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 18$, $BC = 14$, and $PM = 12$, what is $MK$?[[image-1]][[choice-6]]","widgets":{"image-1":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-6":{"type":"choice","options":["$\\dfrac{28}{3}$","$\\dfrac{56}{9}$","$\\dfrac{28}{9}$","$\\dfrac{20}{3}$"],"answer":[0]}},"step":1,"hints":["The ratios of corresponding sides in similar triangles are equal: $\\dfrac{PM}{AB} = \\dfrac{MK}{BC}$.","$\\dfrac{PM}{AB} = \\dfrac{12}{18} = \\dfrac{2}{3}$, so $MK = 14 \\cdot \\dfrac{2}{3} = \\dfrac{28}{3}$."]},{"content":"Two triangles are similar with scale factor $k = 4$ from the smaller to the larger triangle. The smaller triangle has perimeter $18$. What is the perimeter of the larger triangle?[[choice-35]]","widgets":{"choice-35":{"type":"choice","options":["$72$","$22$","$4,5$","$54$"],"answer":[0]}},"step":1,"hints":["The perimeters of similar triangles have the same ratio as their corresponding sides.","If $k=4$, then $P_2 = P_1 \\cdot k = 18 \\cdot 4 = 72$."]},{"content":"Two triangles are similar with scale factor $k = 3$ from the smaller to the larger triangle. The smaller triangle has area $5\\ \\text{cm}^2$. What is the area of the larger triangle?[[choice-74]]","widgets":{"choice-74":{"type":"choice","options":["$10\\ \\text{cm}^2$","$15\\ \\text{cm}^2$","$45\\ \\text{cm}^2$","$30\\ \\text{cm}^2$"],"answer":[2]}},"step":1,"hints":["The areas of similar figures are in the ratio $k^2$.","$S_2 = 5 \\cdot 3^2 = 5 \\cdot 9 = 45\\ \\text{cm}^2$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. An altitude of the smaller triangle is $h_1 = 15$, and the scale factor from smaller to larger is $k = \\dfrac{8}{5}$. Find the corresponding altitude $h$ of the larger triangle.[[image-114]][[choice-130]]","widgets":{"image-114":{"type":"image","url":""},"choice-130":{"type":"choice","options":["$20$","$22$","$24$","$26$"],"answer":[2]}},"step":1,"hints":["All corresponding lengths in similar triangles, including altitudes, medians, and sides, are multiplied by $k$.","$h = 15 \\cdot \\dfrac{8}{5} = 24$."]},{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 25$, $BC = 16$, and $PM = 15$, what is $MK$?[[image-2]][[choice-7]]","widgets":{"image-2":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-7":{"type":"choice","options":["$\\dfrac{48}{5}$","$\\dfrac{36}{5}$","$\\dfrac{45}{8}$","$\\dfrac{32}{5}$"],"answer":[1]}},"step":1,"hints":["Use the proportion between corresponding sides: $\\dfrac{PM}{AB} = \\dfrac{MK}{BC}$.","$\\dfrac{PM}{AB}=\\dfrac{15}{25}=\\dfrac{3}{5}$, so $MK=16\\cdot\\dfrac{3}{5}=\\dfrac{48}{5}$."]},{"content":"Triangles $ABC$ and $PMK$ are similar. Their corresponding sides are $AB \\leftrightarrow PM$, $BC \\leftrightarrow MK$, and $AC \\leftrightarrow PK$. If $AB = 21$, $BC = 30$, and $PM = 14$, what is $MK$?[[image-3]][[choice-8]]","widgets":{"image-3":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2025/09/2-01-22.svg"},"choice-8":{"type":"choice","options":["$18$","$20$","$\\dfrac{70}{3}$","$\\dfrac{40}{3}$"],"answer":[1]}},"step":1,"hints":["Find the scale factor using corresponding sides: $\\dfrac{PM}{AB}$.","$\\dfrac{PM}{AB}=\\dfrac{14}{21}=\\dfrac{2}{3}$, so $MK=30\\cdot\\dfrac{2}{3}=20$."]},{"content":"Two triangles are similar with scale factor $k = 2$ from the smaller to the larger triangle. The smaller triangle has perimeter $27$. What is the perimeter of the larger triangle?[[choice-36]]","widgets":{"choice-36":{"type":"choice","options":["$54$","$29$","$13,5$","$81$"],"answer":[0]}},"step":1,"hints":["The perimeter is multiplied by $k$, just like the side lengths.","$P_2 = P_1\\cdot k = 27\\cdot 2 = 54$."]},{"content":"Two triangles are similar with scale factor $k = \\dfrac{7}{3}$ from the smaller to the larger triangle. The smaller triangle has perimeter $21$. What is the perimeter of the larger triangle?[[choice-37]]","widgets":{"choice-37":{"type":"choice","options":["$49$","$28$","$63$","$10,5$"],"answer":[0]}},"step":1,"hints":["The ratio of the perimeters of similar triangles equals the scale factor $k$.","$P_2 = 21\\cdot\\dfrac{7}{3} = 49$."]},{"content":"Two triangles are similar with scale factor $k = 2$ from the smaller to the larger triangle. The smaller triangle has area $9\\ \\text{cm}^2$. What is the area of the larger triangle?[[choice-75]]","widgets":{"choice-75":{"type":"choice","options":["$18\\ \\text{cm}^2$","$27\\ \\text{cm}^2$","$36\\ \\text{cm}^2$","$45\\ \\text{cm}^2$"],"answer":[2]}},"step":1,"hints":["The areas of similar figures are in the ratio $k^2$.","$S_2 = 9\\cdot 2^2 = 9\\cdot 4 = 36\\ \\text{cm}^2$."]},{"content":"Two triangles are similar with scale factor $k = \\dfrac{3}{4}$ from the smaller to the larger triangle. The larger triangle has area $64\\ \\text{cm}^2$. What is the area of the smaller triangle?[[choice-76]]","widgets":{"choice-76":{"type":"choice","options":["$36\\ \\text{cm}^2$","$48\\ \\text{cm}^2$","$54\\ \\text{cm}^2$","$56\\ \\text{cm}^2$"],"answer":[1]}},"step":1,"hints":["If the length scale factor is $k$, the areas are in the ratio $k^2$.","$S_{\\text{smaller}} = 64\\cdot\\left(\\dfrac{3}{4}\\right)^2 = 64\\cdot\\dfrac{9}{16}=36\\ \\text{cm}^2$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. A median of the smaller triangle is $m_1 = 18$, and the scale factor from smaller to larger is $k = \\dfrac{5}{3}$. Find the corresponding median $m$ of the larger triangle.[[image-115]][[choice-131]]","widgets":{"image-115":{"type":"image","url":""},"choice-131":{"type":"choice","options":["$30$","$24$","$\\dfrac{54}{5}$","$27$"],"answer":[0]}},"step":1,"hints":["Medians are multiplied by the same scale factor as corresponding sides: $k$.","$m = 18\\cdot\\dfrac{5}{3} = 30$."]},{"content":"Triangles $ABC$ and $A_1B_1C_1$ are similar. An angle bisector segment in the smaller triangle has length $l_1 = 9$, and the scale factor from smaller to larger is $k = \\dfrac{13}{6}$. Find the corresponding length $l$ in the larger triangle.[[image-116]][[choice-132]]","widgets":{"image-116":{"type":"image","url":""},"choice-132":{"type":"choice","options":["$\\dfrac{39}{2}$","$\\dfrac{26}{3}$","$18$","$21$"],"answer":[0]}},"step":1,"hints":["All corresponding lengths, including angle bisector segments, are multiplied by $k$.","$l = 9\\cdot\\dfrac{13}{6} = \\dfrac{39}{2}$."]}]}