https://file.aiquickdraw.com/mathbot/file/1727116410648qco0b08u.jpg

The problem in the image involves an isosceles triangle, with a given base of 24.2 cm, an apex angle of 78°, and the goal is to calculate the length of the equal sides (denoted by \( p \)).

We can solve this using trigonometry, specifically the cosine rule. The cosine rule is given by:

\[
p^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

where \( p \) is the length of the side we are calculating, \( a \) and \( b \) are the other sides of the triangle (both equal to \( p \) in this case), and \( C \) is the angle between them, which is 78°. However, a more direct method here, since the triangle is isosceles, is to use right triangle trigonometry by dividing the isosceles triangle into two right-angled triangles.

Each half of the base will be \( 12.1 \) cm (since \( 24.2 / 2 = 12.1 \) cm), and the angle at the top will be split into two 39° angles (half of 78°). 

Now, we can use the trigonometric relationship involving cosine:

\[
\cos(39^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]

Here, the adjacent side is \( 12.1 \, \text{cm} \) and the hypotenuse is \( p \).

Rearranging for \( p \):

\[
p = \frac{12.1}{\cos(39^\circ)}
\]

Let me calculate this.The length of \( p \), the equal sides of the isosceles triangle, is approximately **15.6 cm** to one decimal place.

Would you like more details on this solution, or do you have any other questions?

Here are 5 related questions you might find helpful:
1. How do you solve an isosceles triangle problem when only angles are given?
2. Can the sine rule also be used to solve this problem? How?
3. How would you calculate the area of the isosceles triangle?
4. What changes if the apex angle were larger, say 90°?
5. Can the Pythagorean theorem be applied in isosceles triangles?

**Tip:** For more complex triangles, the cosine and sine rules are extremely useful tools to remember.

A logo for a campsite shows a drawing of a tent in the shape of an isosceles triangle. The base is 24.2 cm, and the apex angle is 78°. Calculate the length p. Give your answer in centimetres to 1 decimal place.

Learn how to calculate the side length of an isosceles triangle given a base of 24.2 cm and an apex angle of 78°. This problem uses key trigonometric concepts, including the cosine rule, and is suitable for students in Grades 6-8.

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