Question

using algebra the two polygons are similar. find the values of x and y. 41. 27 21 39 x - 6 18 y

Explanation:

Step1: Find the scale factor

Since the polygons are similar, the ratios of corresponding sides are equal. First, find the scale factor using the sides 21 and 18. The scale factor from the first polygon to the second is $\frac{18}{21}=\frac{6}{7}$. Wait, no, actually, let's check the corresponding sides. Wait, the first triangle (or quadrilateral? Wait, the figures are triangles? Wait, the first figure has sides 21, 27, 39, and the second has sides $x - 6$, $y$, 18. Wait, maybe the corresponding sides are 21 and 18, 27 and $y$, 39 and $x - 6$? Wait, no, let's see the angles. The marked angles are equal, so the sides opposite or adjacent? Wait, maybe the sides 21 and 18 are corresponding, 27 and $y$ are corresponding, 39 and $x - 6$ are corresponding. So the ratio of similarity is $\frac{18}{21}=\frac{6}{7}$. Wait, no, maybe the first polygon's side 21 corresponds to the second's 18? Wait, no, maybe the other way. Wait, let's do it properly. Let's assume that the first polygon (left) and the second (right) are similar, so the ratio of corresponding sides is equal. Let's take the sides 21 (left) and 18 (right). So the ratio of left to right is $\frac{21}{18}=\frac{7}{6}$. Wait, that makes more sense. So the scale factor from left to right is $\frac{6}{7}$? No, wait, similarity ratio: if two figures are similar, the ratio of their corresponding sides is the scale factor. So if the left figure has a side of length 21, and the right has 18, then the ratio of left to right is $\frac{21}{18}=\frac{7}{6}$, so the right figure is a scaled version of the left by $\frac{6}{7}$? Wait, no, maybe the left is the original, right is the image. So corresponding sides: left side 21 corresponds to right side 18, left side 27 corresponds to right side $y$, left side 39 corresponds to right side $x - 6$. So the ratio of left to right is $\frac{21}{18}=\frac{7}{6}$, so the ratio of right to left is $\frac{6}{7}$. Wait, no, let's set up proportions. For the sides 21 (left) and 18 (right): $\frac{21}{18}=\frac{27}{y}=\frac{39}{x - 6}$. Let's solve for $y$ first. From $\frac{21}{18}=\frac{27}{y}$, cross - multiply: $21y = 18\times27$. Then $y=\frac{18\times27}{21}=\frac{18\times9}{7}=\frac{162}{7}$? Wait, that can't be right. Wait, maybe I mixed up the corresponding sides. Wait, maybe the side 27 (left) corresponds to $y$ (right), and 21 (left) corresponds to 18 (right), and 39 (left) corresponds to $x - 6$ (right). Wait, no, let's look at the angles. The marked angles are equal, so the sides adjacent to the equal angles: in the left figure, the side with length 27 is adjacent to the marked angle, and in the right figure, the side with length $y$ is adjacent to the marked angle. The side with length 21 in the left is opposite? Wait, maybe the correct corresponding sides are 21 (left) and 18 (right), 27 (left) and $y$ (right), 39 (left) and $x - 6$ (right). Wait, let's try again. Let's set the proportion for the sides 21 and 18: $\frac{21}{18}=\frac{27}{y}$. Solving for $y$: $21y = 18\times27$ → $y=\frac{18\times27}{21}=\frac{18\times9}{7}=\frac{162}{7}\approx23.14$? That seems odd. Wait, maybe the other way: $\frac{18}{21}=\frac{y}{27}$. Then $y=\frac{18\times27}{21}=\frac{18\times9}{7}=\frac{162}{7}$ same as before. Now for $x - 6$: $\frac{18}{21}=\frac{x - 6}{39}$. Cross - multiply: $21(x - 6)=18\times39$. Then $21x-126 = 702$. Then $21x=702 + 126=828$. Then $x=\frac{828}{21}=\frac{276}{7}\approx39.43$? That doesn't seem right. Wait, maybe I mixed up the corresponding sides. Let's check the lengths. 21, 27, 39. Let's see if 21, 39, 27 are a triangle? Wait, 21 + 27>39? 48>39, yes. 21 + 39>27, yes. 27 + 39>21, yes. Now the right figure has sides $x - 6$, 18, $y$. Let's assume that the sides 21 and 18 are corresponding, 39 and $x - 6$ are corresponding, 27 and $y$ are corresponding. Wait, maybe the ratio is $\frac{39}{x - 6}=\frac{21}{18}$. So $\frac{39}{x - 6}=\frac{7}{6}$. Cross - multiply: $7(x - 6)=39\times6$. $7x-42 = 234$. $7x=234 + 42=276$. $x=\frac{276}{7}\approx39.43$. Still odd. Wait, maybe the sides 27 and $x - 6$ are corresponding, 21 and 18, 39 and $y$. Let's try that. So $\frac{21}{18}=\frac{39}{y}=\frac{27}{x - 6}$. First, $\frac{21}{18}=\frac{39}{y}$ → $21y=18\times39$ → $y=\frac{18\times39}{21}=\frac{18\times13}{7}=\frac{234}{7}\approx33.43$. Then $\frac{21}{18}=\frac{27}{x - 6}$ → $21(x - 6)=18\times27$ → $21x-126 = 486$ → $21x=612$ → $x=\frac{612}{21}=\frac{204}{7}\approx29.14$. Still not nice numbers. Wait, maybe the problem is that the figures are quadrilaterals? Wait, the left figure looks like a quadrilateral divided into two triangles? Wait, no, the left figure is a triangle? Wait, no, the left figure has three sides: 21, 27, 39, so it's a triangle. The right figure also has three sides: $x - 6$, 18, $y$, so it's a triangle. The marked angles are equal, so they are similar by SAS similarity? Wait, the included angle is equal, and the sides around the angle are proportional. Wait, maybe the sides adjacent to the marked angle: in the left triangle, the sides adjacent to the marked angle are 27 and 39? No, the marked angle is at the top. Wait, maybe the left triangle has sides 27 (adjacent to angle), 39 (opposite?), no. Wait, maybe the correct corresponding sides are 21 and 18, 27 and $y$, 39 and $x - 6$, and the ratio is $\frac{21}{18}=\frac{7}{6}$, so the scale factor is $\frac{6}{7}$ (from left to right). Wait, no, similarity ratio is the ratio of corresponding sides. If left has side 21, right has 18, then the ratio of left to right is $\frac{21}{18}=\frac{7}{6}$, so the right triangle is smaller by a factor of $\frac{6}{7}$. So the side 27 on the left corresponds to $y$ on the right: $y = 27\times\frac{6}{7}=\frac{162}{7}\approx23.14$. The side 39 on the left corresponds to $x - 6$ on the right: $x - 6=39\times\frac{6}{7}=\frac{234}{7}\approx33.43$, so $x=\frac{234}{7}+6=\frac{234 + 42}{7}=\frac{276}{7}\approx39.43$. But these are fractions. Maybe I made a mistake in identifying corresponding sides. Wait, maybe the sides 21 and 18 are not corresponding. Let's look at the lengths again. 21, 27, 39. Let's see if 21 and 39 are a ratio of 7:13 (21 = 3×7, 39 = 3×13), 27 is 3×9. 18 is 3×6. Oh! Wait a minute. 21 = 3×7, 18 = 3×6, 27 = 3×9, 39 = 3×13. Ah! So the sides are multiples of 3. So 21 = 3×7, 18 = 3×6, 27 = 3×9, 39 = 3×13. So the ratio of the left triangle's sides (7, 9, 13) to the right triangle's sides (6, $y/3$, $(x - 6)/3$) should be 7:6. Wait, because 7 (from 21/3) corresponds to 6 (from 18/3). So 7:6. So 9 (from 27/3) corresponds to $y/3$, so $\frac{7}{6}=\frac{9}{y/3}$? No, wait, the right triangle's sides (divided by 3) are 6, $y/3$, $(x - 6)/3$. The left triangle's sides (divided by 3) are 7, 9, 13. So the ratio of left to right is 7:6. So 7 corresponds to 6, 9 corresponds to $y/3$, 13 corresponds to $(x - 6)/3$. So $\frac{7}{6}=\frac{9}{y/3}$ → $\frac{7}{6}=\frac{27}{y}$ → $7y = 162$ → $y=\frac{162}{7}$? No, that's not matching. Wait, no, maybe the right triangle's sides are 6, $(x - 6)/3$, $y/3$, and the left is 7, 13, 9. So 7 corresponds to 6, 13 corresponds to $(x - 6)/3$, 9 corresponds to $y/3$. So $\frac{7}{6}=\frac{13}{(x - 6)/3}$ → $\frac{7}{6}=\frac{39}{x - 6}$ → $7(x - 6)=234$ → $7x-42 = 234$ → $7x=276$ → $x=\frac{276}{7}\approx39.43$. And $\frac{7}{6}=\frac{9}{y/3}$ → $\frac{7}{6}=\frac{27}{y}$ → $7y = 162$ → $y=\frac{162}{7}\approx23.14$. But these are not integers. Maybe the problem has a typo, or I misidentified the corresponding sides. Wait, maybe the sides 21 and 18 are corresponding, 27 and $x - 6$ are corresponding, 39 and $y$ are corresponding. Let's try that. So $\frac{21}{18}=\frac{27}{x - 6}=\frac{39}{y}$. First, $\frac{21}{18}=\frac{27}{x - 6}$ → $21(x - 6)=18\times27$ → $21x-126 = 486$ → $21x=612$ → $x=\frac{612}{21}=\frac{204}{7}\approx29.14$. $\frac{21}{18}=\frac{39}{y}$ → $21y=18\times39$ → $y=\frac{702}{21}=\frac{234}{7}\approx33.43$. Still fractions. Wait, maybe the original figures are quadrilaterals, not triangles. Let's assume that the left figure is a quadrilateral with sides 21, 27, 39, and another side (let's say the fourth side is, but it's not shown). Wait, the diagram shows two triangles? No, the left figure looks like a quadrilateral split into two triangles by a diagonal. Wait, maybe the left quadrilateral has sides 21, 27, and the diagonal 39, and the right quadrilateral has sides 18, $y$, and the diagonal $x - 6$. So the two triangles formed by the diagonal are similar. So the triangle with sides 21, 27, 39 is similar to the triangle with sides 18, $y$, $x - 6$. So using the Law of Sines or Law of Cosines, but since they are similar, the ratios of sides are equal. Wait, maybe the sides 21 and 18 are corresponding, 27 and $y$ are corresponding, 39 and $x - 6$ are corresponding. So $\frac{21}{18}=\frac{27}{y}=\frac{39}{x - 6}$. Let's simplify $\frac{21}{18}=\frac{7}{6}$. So $\frac{7}{6}=\frac{27}{y}$ → $y=\frac{27\times6}{7}=\frac{162}{7}\approx23.14$, and $\frac{7}{6}=\frac{39}{x - 6}$ → $x - 6=\frac{39\times6}{7}=\frac{234}{7}\approx33.43$, so $x=\frac{234}{7}+6=\frac{234 + 42}{7}=\frac{276}{7}\approx39.43$. But these are not nice numbers. Maybe there's a mistake in my assumption. Wait, maybe the corresponding sides are 27 and 18, 21 and $y$, 39 and $x - 6$. Let's try that. So $\frac{27}{18}=\frac{3}{2}$. Then $\frac{21}{y}=\frac{3}{2}$ → $3y = 42$ → $y = 14$. And $\frac{39}{x - 6}=\frac{3}{2}$ → $3(x - 6)=78$ → $x - 6 = 26$ → $x = 32$. Ah! This gives integer values. So maybe I had the corresponding sides wrong. Let's check: 27 (left) corresponds to 18 (right), ratio $\frac{27}{18}=\frac{3}{2}$. Then 21 (left) corresponds to $y$ (right): $\frac{21}{y}=\frac{3}{2}$ → $y = 14$. 39 (left) corresponds to $x - 6$ (right): $\frac{39}{x - 6}=\frac{3}{2}$ → $x - 6 = 26$ → $x = 32$. Let's verify the ratios: $\frac{27}{18}=\frac{3}{2}$, $\frac{21}{14}=\frac{3}{2}$, $\frac{39}{26}=\frac{3}{2}$. Yes! All ratios are $\frac{3}{2}$. So that works. So I must have mixed up the corresponding sides earlier. The side 27 (left) corresponds to 18 (right), 21 (left) corresponds to $y$ (right), and 39 (left) corresponds to $x - 6$ (right).

Step2: Solve for y

Using the ratio $\frac{21}{y}=\frac{3}{2}$ (since $\frac{27}{18}=\frac{3}{2}$ is the similarity ratio). Cross - multiply: $3y=21\times2$ → $3y = 42$ → $y = 14$.

Step3: Solve for x

Using the ratio $\frac{39}{x - 6}=\frac{3}{2}$. Cross - multiply: $3(x - 6)=39\times2$ → $3x-18 = 78$ → $3x=78 + 18=96$ → $x=\frac{96}{3}=32$.

Answer:

$x = 32$, $y = 14$