Question

vincent started with a triangle with a base of 2 inches and a height of 5 inches. he wants to enlarge the triangle so that the new base is 9 inches. which proportion is correctly set up to find the height of the enlarged triangle?
\\( \frac { 2 } { 9 } = \frac { 5 } { x } \\)
\\( \frac { 5 } { x } = \frac { 9 } { 2 } \\)
\\( \frac { 9 } { 5 } = \frac { x } { 2 } \\)
\\( \frac { 9 } { 2 } = \frac { x } { 5 } \\)

Explanation:

Step1: Understand the proportion concept

In similar triangles, the ratio of corresponding sides (base and height) is equal. The original triangle has base \(2\) inches and height \(5\) inches. The new base is \(9\) inches, and let the new height be \(x\) inches.

Step2: Set up the proportion

The ratio of base to height for the original triangle is \(\frac{2}{5}\), and for the new triangle is \(\frac{9}{x}\). Since the triangles are similar, \(\frac{2}{5}=\frac{9}{x}\) is incorrect. The correct proportion should be \(\frac{2}{9}=\frac{5}{x}\) (ratio of original base to new base equals ratio of original height to new height) or \(\frac{9}{2}=\frac{x}{5}\) (cross - multiplying gives the same relationship as \(\frac{2}{5}=\frac{9}{x}\) is wrong, \(\frac{9}{5}=\frac{2}{x}\) is wrong, \(\frac{2}{x}=\frac{5}{9}\) is wrong).

Answer:

\(\frac{2}{9}=\frac{5}{x}\) (or \(\frac{9}{2}=\frac{x}{5}\))