Question

congruence, similarity, and symmetry
example the two triangles are similar. calculate the missing sides in the smaller triangle.
corresponding sides: 9 15 18 larger triangle
6 x y smaller triangle
\\( \frac { 15 } { x } = \frac { 9 } { 6 } \\)
and
\\( \frac { 18 } { y } = \frac { 9 } { 6 } \\)
\\( ( 15 ) ( 6 ) = 9 x \\)
(set cross products)
\\( ( 18 ) ( 6 ) = 9 y \\)
\\( 90 = 9 x \\)
(divide by 9.)
\\( 108 = 9 y \\)
\\( 10 = x \\)
\\( 12 = y \\)
directions given similar triangles, find the missing sides.

1. find the length of side x. _
2. find the length of side y. _
3. find the length of side x. _
4. find the length of side y. _
5. find the length of side x. _
6. find the length of side y. _
7. find the length of side x. _
8. find the length of side y. _
9. find the length of side x. _
10. find the length of side y. _

Explanation:

Step1: Set up proportion for similar triangles

For similar triangles, the ratios of corresponding sides are equal. For problem 1: \(\frac{4}{8}=\frac{10}{x}\)

Step2: Cross - multiply

\(4x = 8\times10\)

Step3: Solve for \(x\)

\(4x=80\), then \(x = 20\)

For problem 2: \(\frac{4}{8}=\frac{8}{y}\)

Step1: Set up proportion

\(\frac{4}{8}=\frac{8}{y}\)

Step2: Cross - multiply

\(4y=8\times8\)

Step3: Solve for \(y\)

\(4y = 64\), then \(y = 16\)

For problem 3: \(\frac{12}{9}=\frac{16}{x}\)

Step1: Set up proportion

\(\frac{12}{9}=\frac{16}{x}\)

Step2: Cross - multiply

\(12x=9\times16\)

Step3: Solve for \(x\)

\(12x = 144\), then \(x = 12\)

For problem 4: \(\frac{12}{9}=\frac{24}{y}\)

Step1: Set up proportion

\(\frac{12}{9}=\frac{24}{y}\)

Step2: Cross - multiply

\(12y=9\times24\)

Step3: Solve for \(y\)

\(12y = 216\), then \(y = 18\)

For problem 5: \(\frac{7}{28}=\frac{13}{x}\)

Step1: Set up proportion

\(\frac{7}{28}=\frac{13}{x}\)

Step2: Cross - multiply

\(7x=28\times13\)

Step3: Solve for \(x\)

\(7x = 364\), then \(x = 52\)

For problem 6: \(\frac{7}{28}=\frac{16}{y}\)

Step1: Set up proportion

\(\frac{7}{28}=\frac{16}{y}\)

Step2: Cross - multiply

\(7y=28\times16\)

Step3: Solve for \(y\)

\(7y = 448\), then \(y = 64\)

For problem 7: \(\frac{3}{y}=\frac{6}{x}=\frac{5}{15}\)
First, from \(\frac{5}{15}=\frac{1}{3}\), then from \(\frac{3}{y}=\frac{1}{3}\), \(y = 9\); from \(\frac{6}{x}=\frac{1}{3}\), \(x = 18\)

For problem 8: Using the same proportion as above (from problem 7), \(y = 9\)

For problem 9: \(\frac{6}{x}=\frac{12}{16}\)

Step1: Set up proportion

\(\frac{6}{x}=\frac{12}{16}\)

Step2: Cross - multiply

\(12x=6\times16\)

Step3: Solve for \(x\)

\(12x = 96\), then \(x = 8\)

For problem 10: \(\frac{12}{16}=\frac{15}{y}\)

Step1: Set up proportion

\(\frac{12}{16}=\frac{15}{y}\)

Step2: Cross - multiply

\(12y=16\times15\)

Step3: Solve for \(y\)

\(12y = 240\), then \(y = 20\)

Answer:

1. \(20\)
2. \(16\)
3. \(12\)
4. \(18\)
5. \(52\)
6. \(64\)
7. \(18\)
8. \(9\)
9. \(8\)
10. \(20\)