Question

introduction to similarity (tues 1/20)
find the missing the side.
10
8
?
4
14
7

Explanation:

Step1: Determine the scale factor

The corresponding sides of similar triangles are proportional. Let's find the scale factor using the known sides. The side of length 8 in the first triangle corresponds to the side of length 4 in the second triangle. So the scale factor $k$ is $\frac{4}{8} = \frac{1}{2}$. We can also check with the base: the base of the first triangle is 14 and the base of the second is 7, and $\frac{7}{14} = \frac{1}{2}$, so the scale factor is $\frac{1}{2}$.

Step2: Find the missing side

The side of length 10 in the first triangle corresponds to the missing side (let's call it $x$) in the second triangle. Using the scale factor, we have $x = 10\times\frac{1}{2}$? Wait, no, wait. Wait, actually, the scale factor from the first triangle to the second is $\frac{1}{2}$, so the second triangle is smaller. Wait, alternatively, we can set up a proportion. Let the missing side be $x$. Then $\frac{x}{10} = \frac{4}{8} = \frac{7}{14}$. Let's use $\frac{x}{10} = \frac{4}{8}$. Simplify $\frac{4}{8} = \frac{1}{2}$, so $x = 10\times\frac{1}{2}$? Wait, no, that would be 5, but wait, let's check with the base. $\frac{7}{14} = \frac{1}{2}$, and $\frac{4}{8} = \frac{1}{2}$, so the scale factor is $\frac{1}{2}$. So the side corresponding to 10 in the first triangle is $10\times\frac{1}{2}$? Wait, no, wait, maybe I mixed up the triangles. Wait, the first triangle has sides 10, 8, 14. The second has sides $x$, 4, 7. So the ratio of corresponding sides should be equal. So $\frac{x}{10} = \frac{4}{8} = \frac{7}{14}$. Let's take $\frac{4}{8} = \frac{1}{2}$, and $\frac{7}{14} = \frac{1}{2}$, so the ratio is $\frac{1}{2}$ from first to second? Wait, no, actually, the second triangle is a scaled-down version of the first. So the ratio of second to first is $\frac{1}{2}$. So $x = 10\times\frac{1}{2}$? Wait, no, that would be 5, but let's check: if $x = 5$, then the sides of the second triangle are 5, 4, 7. Let's check the ratios: $\frac{5}{10} = \frac{1}{2}$, $\frac{4}{8} = \frac{1}{2}$, $\frac{7}{14} = \frac{1}{2}$. Yes, that works. Wait, but alternatively, let's set up the proportion correctly. Let's say the two triangles are similar, so the ratio of corresponding sides is equal. So $\frac{x}{10} = \frac{4}{8}$. Cross-multiplying, $8x = 10\times4$, so $8x = 40$, so $x = 5$. Alternatively, $\frac{x}{10} = \frac{7}{14}$, so $\frac{x}{10} = \frac{1}{2}$, so $x = 5$. Yep, that's correct.

Wait, let's do it properly. Let's denote the first triangle as Triangle A with sides 10, 8, 14. Triangle B with sides $x$, 4, 7. Since they are similar, the ratios of corresponding sides are equal. So $\frac{x}{10} = \frac{4}{8} = \frac{7}{14}$. Let's take $\frac{4}{8} = \frac{1}{2}$, so $\frac{x}{10} = \frac{1}{2}$, so $x = 5$. Or $\frac{7}{14} = \frac{1}{2}$, so $\frac{x}{10} = \frac{1}{2}$, so $x = 5$. So the missing side is 5.

Answer:

5