In ∆TAP, ∠TAP = 60°, TA = 6 cm, AP = 8 cm. K is the midpoint of AP. A line from K is
produced to meet TP at O such that ∠AKO = 120°. Find the length of OK.
Ans 1. 3 cm
2. 4 cm
3. 5 cm
4. 6 cm


Given the triangle \( \Delta TAP \) with \( \angle TAP = 60^\circ \), \( TA = 6 \) cm, and \( AP = 8 \) cm. \( K \) is the midpoint of \( AP \). A line from \( K \) is produced to meet \( TP \) at \( O \) such that \( \angle AKO = 120^\circ \). We need to find the length of \( OK \).

First, let's break down the problem and use geometric properties to solve it.

1. Since \( K \) is the midpoint of \( AP \), we have:
   \[ AK = KP = \frac{AP}{2} = \frac{8}{2} = 4 \text{ cm} \]

2. Given \( \angle AKO = 120^\circ \), we can use the law of cosines in triangle \( AKO \):
   \[
   OK^2 = AK^2 + KO^2 - 2 \cdot AK \cdot KO \cdot \cos(120^\circ)
   \]

Since \( AK = 4 \) cm and \( \cos(120^\circ) = -\frac{1}{2} \), we get:
\[ OK^2 = 4^2 + KO^2 - 2 \cdot 4 \cdot KO \cdot \left(-\frac{1}{2}\right) \]
\[ OK^2 = 16 + KO^2 + 4KO \]

Now, let's consider the length \( KO \). Since \( K \) is the midpoint and forms \( \angle AKO = 120^\circ \), \( OK \) actually equals \( AK \).

Thus, the length \( OK = AK = 4 \text{ cm} \).

Therefore, the correct answer is:

**2. 4 cm**

Would you like further details or have any questions?

Here are 5 questions you might want to ask:
1. How do you apply the law of cosines in solving triangles?
2. What are the properties of a triangle's midpoints?
3. Can you explain more examples involving geometric constructions?
4. How do you solve problems involving angles greater than 90 degrees?
5. What are some tips for visualizing geometric problems?

Tip: In geometry problems, drawing a diagram and identifying key points and angles can greatly help in understanding and solving the problem efficiently.

Learn how to find the length of OK in triangle TAP where angle TAP is 60 degrees, TA = 6 cm, and AP = 8 cm. K is the midpoint of AP, and a line from K forms angle AKO = 120 degrees with TP. Step-by-step solution using geometric properties and the law of cosines. Suitable for high school students.

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