Find the smallest angle of an isosceles triangle that has one side of length 2.9cm and two sides of length 7.4cm.

[Hint: Draw a sketch of the isosceles triangle then sketch its line of symmetry to split the isosceles triangle into two right-angled triangles.] to 3 significant figures

To find the smallest angle of an isosceles triangle where one side is 2.9 cm and the two equal sides are 7.4 cm, let's follow the given hint:

### Step-by-step approach:

1. **Draw the triangle**: 
   Label the isosceles triangle with vertices \(A\), \(B\), and \(C\), where \(AB = AC = 7.4 \, \text{cm}\) and \(BC = 2.9 \, \text{cm}\).

2. **Line of symmetry**: 
   The line of symmetry splits the triangle into two right-angled triangles. Each half of the base \(BC\) will measure:
   \[
   \frac{BC}{2} = \frac{2.9}{2} = 1.45 \, \text{cm}
   \]
   Let the point where the line of symmetry meets \(BC\) be \(D\). Then, \(BD = DC = 1.45 \, \text{cm}\).

3. **Use trigonometry**: 
   In one of the right-angled triangles, say \(ABD\), we know:
   - Hypotenuse \(AB = 7.4 \, \text{cm}\)
   - Opposite side \(BD = 1.45 \, \text{cm}\)

   The angle we're trying to find is \(\angle ADB\), half of the apex angle of the isosceles triangle. Using the sine function:
   \[
   \sin(\angle ADB) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1.45}{7.4}
   \]
   Now calculate this:
   \[
   \sin(\angle ADB) = \frac{1.45}{7.4} \approx 0.196
   \]

4. **Find the angle**:
   Now, calculate \(\angle ADB\) using the inverse sine (\(\sin^{-1}\)):
   \[
   \angle ADB = \sin^{-1}(0.196) \approx 11.3^\circ
   \]
   Therefore, the apex angle (the angle opposite the base of the isosceles triangle) is:
   \[
   \angle A = 2 \times 11.3^\circ = 22.6^\circ
   \]

5. **Smallest angle**: 
   The two equal base angles in the triangle are the remaining angles. Since the sum of angles in a triangle is \(180^\circ\):
   \[
   2 \times \text{base angle} + 22.6^\circ = 180^\circ
   \]
   \[
   2 \times \text{base angle} = 180^\circ - 22.6^\circ = 157.4^\circ
   \]
   \[
   \text{Base angle} = \frac{157.4^\circ}{2} = 78.7^\circ
   \]
   Hence, the smallest angle is the apex angle, which is \(22.6^\circ\) to 3 significant figures.

### Summary:
The smallest angle of the isosceles triangle is \(22.6^\circ\).

Would you like more details or have any further questions?

Here are 5 related questions:
1. How would the smallest angle change if the base was longer?
2. Can you calculate the height of the triangle?
3. What is the perimeter of this triangle?
4. How can you verify your answer using the cosine rule?
5. What is the area of this triangle?

**Tip**: When dealing with isosceles triangles, symmetry can simplify calculations significantly by splitting the triangle into two right-angled triangles.

Find the smallest angle of an isosceles triangle that has one side of length 2.9cm and two sides of length 7.4cm.

Learn how to find the smallest angle in an isosceles triangle with side lengths of 7.4 cm and 2.9 cm. This solution uses trigonometric ratios and the angle sum theorem, with a detailed step-by-step approach. Suitable for students in Grades 8-10.

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