Question

part 2 of 4
triangles abc and dbe are similar. therefore the ratios of the corresponding sides are equal. complete the equation relating the corresponding sides.
\\(\frac{ab}{bd} = \frac{bc}{be}\\) (with be checked)

part 3 of 4
substitute 7 for bd, 21 for ab and 8 for be and then solve for bc.
\\(\frac{ab}{bd} = \frac{bc}{be}\\)
\\(\frac{21}{7} = \frac{bc}{8}\\)
\\(21 \times (8) = bc\\) (wait, correction: cross - multiply gives \\(21\times8 = 7\times bc\\), but the original ocr has \\(21\times(8)=bc\\) which might be a typo, but we follow ocr)
blank = bc

Explanation:

Step1: Substitute values into the proportion

We know the proportion is \(\frac{AB}{BD}=\frac{BC}{BE}\). Given \(AB = 21\), \(BD = 7\), and \(BE = 8\), substitute these values into the left - hand side of the proportion. So \(\frac{21}{7}=\frac{BC}{8}\).

Step2: Simplify the left - hand side

Simplify \(\frac{21}{7}\), since \(21\div7 = 3\), the equation becomes \(3=\frac{BC}{8}\).

Step3: Solve for \(BC\)

To solve for \(BC\), we can multiply both sides of the equation \(3=\frac{BC}{8}\) by \(8\). Using the property of equality \(a=\frac{b}{c}\Rightarrow b = a\times c\) (where \(a = 3\), \(b = BC\), and \(c = 8\)), we get \(BC=3\times8 = 24\). Also, for the step where we cross - multiply, from \(\frac{21}{7}=\frac{BC}{8}\), cross - multiplying gives \(21\times8=7\times BC\), but we can also first simplify \(\frac{21}{7}\) to 3. Then \(3\times8 = BC\), so \(BC = 24\). Filling in the blanks: The first blank (after 21) is \(\frac{21}{7}\) (which simplifies to 3), the second blank (the numerator when cross - multiplying or the coefficient) is 7 (wait, no, let's re - examine. The first blank in the first box (the fraction) is \(\frac{21}{7}\), the second blank (the denominator when cross - multiplying? Wait, the original equation after substitution is \(\frac{21}{7}=\frac{BC}{8}\). Then cross - multiplying: \(21\times8=7\times BC\), but if we do \(\frac{21}{7}\times8 = BC\), \(\frac{21}{7}=3\), so \(3\times8 = BC\). So the first blank (the fraction) is \(\frac{21}{7}\) (or 3), the second blank (the denominator in the cross - multiply step? Wait, the problem shows:

\(\frac{21}{\underline{}}=\frac{BC}{8}\)

Then \(\frac{21}{\underline{}}\times(8)=BC\)

Then \(\underline{} = BC\)

So the first blank (the denominator of the left - hand fraction) is 7, because \(\frac{21}{7}=\frac{BC}{8}\). Then \(\frac{21}{7}\times8=BC\), \(\frac{21}{7}=3\), so \(3\times8 = 24\). So:

First blank (denominator of left fraction): 7

Second blank (the multiplier, which is 8? Wait, no, the first box is \(\frac{21}{\underline{}}\), so that's 7. Then the second box is \(\frac{21}{7}\times\underline{} = BC\), that's 8. Then the third box is 24.

Wait, let's do it step by step:

1. Substitute \(AB = 21\), \(BD = 7\), \(BE = 8\) into \(\frac{AB}{BD}=\frac{BC}{BE}\), we get \(\frac{21}{7}=\frac{BC}{8}\). So the first blank (the denominator of the left - hand side) is 7.

2. To solve for \(BC\), we can multiply both sides by 8: \(\frac{21}{7}\times8=BC\). So the second blank (the number we multiply by) is 8.

3. Then \(\frac{21}{7}=3\), \(3\times8 = 24\), so the third blank is 24.

So:

First box: 7

Second box: 8

Third box: 24

Wait, let's check again. The first equation after substitution is \(\frac{21}{\underline{}}=\frac{BC}{8}\). Since \(BD = 7\), the denominator is 7. Then, to solve for \(BC\), we cross - multiply: \(21\times8=\underline{}\times BC\)? No, wait, the proportion is \(\frac{AB}{BD}=\frac{BC}{BE}\), so \(AB\times BE=BD\times BC\). So \(21\times8 = 7\times BC\). Then \(BC=\frac{21\times8}{7}\). \(21\div7 = 3\), \(3\times8 = 24\). So:

The first blank (denominator of left fraction): 7

The second blank (the denominator in the cross - product, but in the step \(\frac{21}{7}\times8 = BC\), the 8 is from \(BE\), so the second blank is 7? No, I think I messed up. Let's start over.

Given \(\frac{AB}{BD}=\frac{BC}{BE}\), \(AB = 21\), \(BD = 7\), \(BE = 8\).

So \(\frac{21}{7}=\frac{BC}{8}\)

So the first box (the denominator of the left fraction) is 7.

Then, to isolate \(BC\), we multiply both sides by 8: \(\frac{21}{7}\times8=BC\)

\(\frac{21}{7}=3\), so \(3\times8 = 24\), so \(BC = 24\)

So the first blank (denominator) is 7, the second blank (the multiplier) is 8, the third blank is 24.

So:

First box: 7

Second box: 8

Third box: 24

Answer:

First blank (denominator of \(\frac{21}{\underline{}}\)): 7; Second blank (the number multiplied by \(\frac{21}{7}\)): 8; Third blank (value of \(BC\)): 24