Question

consider the enlargement of the triangle. which proportional statements are true for finding the value of x? choose three correct answers. $$ \frac { x } { 9 } = \frac { 9 } { 16 } $$ $$ \frac { 16 } { 6 } = \frac { x } { 9 } $$ $$ \frac { x } { 9 } = \frac { 16 } { 6 } $$ $$ \frac { x } { 6 } = \frac { 16 } { 9 } $$

Explanation:

Step1: Use the property of similar triangles

For similar triangles, the ratios of corresponding sides are equal.
Let the first triangle have sides \(6\) and \(9\), and the second triangle have sides \(16\) and \(x\).
We can set up the proportion in different valid ways.

Step2: Analyze the first proportion

If we consider \(\frac{x}{9}=\frac{16}{6}\), cross - multiplying gives \(6x = 16\times9\). This is a valid proportion as it equates the ratios of corresponding sides.

Step3: Analyze the second proportion

If we consider \(\frac{16}{6}=\frac{x}{9}\), cross - multiplying gives \(6x = 16\times9\). This is also a valid proportion as it just swaps the numerator and denominator of the equal ratios.

Step4: Analyze the third proportion

If we consider \(\frac{x}{16}=\frac{9}{6}\), cross - multiplying gives \(6x = 16\times9\). This is a valid proportion as it equates the ratios of corresponding sides in another order.

Answer:

\(\frac{x}{9}=\frac{16}{6}\), \(\frac{16}{6}=\frac{x}{9}\), \(\frac{x}{16}=\frac{9}{6}\)