how long is the minute hand of a clock if it is 7 cm longer than the hour hand, and the distsnce between the tips of the hands at 9 o clock is 13 cm

Let's denote the length of the hour hand as \( h \) cm and the length of the minute hand as \( m \) cm. According to the problem, the minute hand is 7 cm longer than the hour hand:

\[
m = h + 7
\]

At 9 o'clock, the hour hand and the minute hand are perpendicular to each other, forming a right triangle where the hour hand and the minute hand are the legs, and the distance between their tips (13 cm) is the hypotenuse. According to the Pythagorean theorem:

\[
h^2 + m^2 = 13^2
\]

Substituting \( m = h + 7 \) into the equation:

\[
h^2 + (h + 7)^2 = 169
\]

Now, let's expand and simplify this equation:

\[
h^2 + (h^2 + 14h + 49) = 169
\]
\[
2h^2 + 14h + 49 = 169
\]

Subtract 169 from both sides:

\[
2h^2 + 14h - 120 = 0
\]

Now, divide the entire equation by 2:

\[
h^2 + 7h - 60 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula:

\[
h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For our equation, \( a = 1 \), \( b = 7 \), and \( c = -60 \). Plugging in the values:

\[
h = \frac{-7 \pm \sqrt{7^2 + 240}}{2}
\]
\[
h = \frac{-7 \pm \sqrt{289}}{2}
\]
\[
h = \frac{-7 \pm 17}{2}
\]

This gives us two possible solutions for \( h \):

\[
h = \frac{10}{2} = 5 \quad \text{or} \quad h = \frac{-24}{2} = -12
\]

Since \( h \) must be positive, \( h = 5 \) cm.

Now, using \( m = h + 7 \):

\[
m = 5 + 7 = 12 \text{ cm}
\]

Thus, the length of the minute hand is **12 cm**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How would the problem change if the distance between the tips was different?
2. What is the formula for calculating the distance between the tips at any other time?
3. How would you solve the problem if the lengths of both hands were unknown?
4. What would the hour hand length be if the minute hand were 5 cm longer?
5. What other configurations lead to a right triangle between the clock hands?
6. How does the minute hand length affect the speed of the clock?
7. Can you generalize the problem to solve for any time?
8. How does the Pythagorean theorem apply to other time intervals?

**Tip:** Always check the feasibility of your solutions in the context of the problem to avoid unrealistic results.

Learn how to calculate the length of the minute hand of a clock using the given problem statement. Understand the application of the Pythagorean theorem and quadratic equations in solving distances between clock hands at specific times. Suitable for students in Grades 7-9.

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