Question

1. a = 9 b = □ c = 36

Explanation:

Assuming this is a problem related to the Pythagorean theorem (since \(a\), \(b\), \(c\) are often used for sides of a right triangle, with \(c\) as the hypotenuse), we can use the formula \(a^{2}+b^{2}=c^{2}\) (if \(c\) is the hypotenuse) or other combinations depending on the triangle type. Here we'll assume \(c\) is the hypotenuse.

Step1: Recall the Pythagorean theorem

The Pythagorean theorem for a right - triangle is \(a^{2}+b^{2}=c^{2}\), where \(a\) and \(b\) are the legs and \(c\) is the hypotenuse. We know \(a = 9\) and \(c=36\), and we want to find \(b\). Rearranging the formula for \(b\), we get \(b=\sqrt{c^{2}-a^{2}}\) (assuming \(c\) is the hypotenuse, so \(c > a\) and \(c>b\)).

Step2: Substitute the values of \(a\) and \(c\)

First, calculate \(a^{2}\) and \(c^{2}\). \(a^{2}=9^{2}=81\) and \(c^{2}=36^{2} = 1296\). Then, \(c^{2}-a^{2}=1296 - 81=1215\).

Step3: Calculate the square root of \(1215\)

\(b=\sqrt{1215}=\sqrt{81\times15}=9\sqrt{15}\approx9\times3.873 = 34.857\)

If we assume that \(c\) is a leg and \(b\) is the hypotenuse (since \(36>9\), this is also a valid assumption), the formula would be \(a^{2}+c^{2}=b^{2}\). Then:

Step1: Recall the Pythagorean theorem (alternative case)

Using \(a^{2}+c^{2}=b^{2}\), with \(a = 9\) and \(c = 36\).

Step2: Substitute the values

\(a^{2}=81\), \(c^{2}=1296\), so \(b^{2}=81 + 1296=1377\)

Step3: Calculate the square root of \(1377\)

\(b=\sqrt{1377}=\sqrt{9\times153}=\sqrt{9\times9\times17}=9\sqrt{17}\approx9\times4.123=37.107\)

Since the problem doesn't specify the type of triangle, but given that \(36\) is much larger than \(9\), the more likely case is that \(c\) is the hypotenuse. If we assume a different relationship (for example, if it's a proportion or a linear relationship, but the use of \(a\), \(b\), \(c\) suggests a triangle - related problem, most likely Pythagorean theorem).

If we assume a direct proportion (for example, if \(a\), \(b\), \(c\) are in proportion, but that's less likely with the notation). Another way, if it's a simple linear relationship like \(a\times k=b\) and \(b\times k = c\) (geometric sequence), then \(k=\frac{c}{b}=\frac{b}{a}\), so \(b^{2}=a\times c\). Let's check this case:

Step1: Assume a geometric sequence relationship

If \(a\), \(b\), \(c\) are in geometric sequence, then \(b^{2}=a\times c\)

Step2: Substitute \(a = 9\) and \(c = 36\)

\(b^{2}=9\times36 = 324\)

Step3: Calculate \(b\)

\(b=\sqrt{324}=18\)

Since \(9\), \(18\), \(36\) is a geometric sequence with a common ratio of \(2\), this is a very clean solution. Given that the problem has integer values (likely expected), the geometric sequence assumption is more probable. So we'll go with this.

(geometric sequence case):

Step1: Assume geometric sequence

For a geometric sequence, the square of the middle term ( \(b\)) is equal to the product of the first (\(a\)) and the third (\(c\)) terms, i.e., \(b^{2}=a\times c\)

Step2: Substitute \(a = 9\) and \(c = 36\)

\(b^{2}=9\times36\)

Step3: Solve for \(b\)

\(b^{2}=324\), so \(b = 18\) (we take the positive value since we are dealing with lengths in a geometric or triangle context)

Answer:

If we assume a geometric sequence relationship (which gives an integer answer, likely the intended case), \(b = 18\). If we assume the Pythagorean theorem with \(c\) as hypotenuse, \(b\approx34.86\) (or \(9\sqrt{15}\)), and with \(b\) as hypotenuse, \(b\approx37.11\) (or \(9\sqrt{17}\)). The most probable answer (with integer result) is \(18\).