https://file.aiquickdraw.com/mathbot/file/1729023661739ow26kshw.jpg

This image represents an isosceles triangle where two sides \(a\) and \(b\) are equal, the base is 76 cm, and the vertex angle is \(50^\circ\).

### Step 1: Calculate the other angles
Since it's an isosceles triangle, the two base angles are equal. The sum of the angles in any triangle is \(180^\circ\). So, the sum of the two base angles is:

\[
180^\circ - 50^\circ = 130^\circ
\]

Each base angle is:

\[
\frac{130^\circ}{2} = 65^\circ
\]

### Step 2: Apply the Law of Sines
Using the Law of Sines to find \(a\) and \(b\). The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{76 \text{ cm}}{\sin C}
\]

Where:
- \(A = 65^\circ\),
- \(B = 65^\circ\),
- \(C = 50^\circ\), and
- the base \(76\) cm corresponds to the angle \(C = 50^\circ\).

So:

\[
\frac{a}{\sin 65^\circ} = \frac{76 \text{ cm}}{\sin 50^\circ}
\]

### Step 3: Solve for \(a\) (or \(b\), since they are equal)

\[
a = \frac{76 \text{ cm} \cdot \sin 65^\circ}{\sin 50^\circ}
\]

Using sine values:
- \(\sin 65^\circ \approx 0.9063\),
- \(\sin 50^\circ \approx 0.7660\),

Substituting into the equation:

\[
a = \frac{76 \times 0.9063}{0.7660} \approx 89.91 \text{ cm}
\]

So, \(a \approx 89.91\) cm, and since \(b = a\), \(b \approx 89.91\) cm as well.

### Final Answer:
- \(a \approx 89.91\) cm,
- \(b \approx 89.91\) cm.

Let me know if you'd like more details or further clarifications!

#### Relative Questions:
1. How is the Law of Sines applied to non-right triangles like this one?
2. How would you use the Law of Cosines to solve a triangle problem?
3. What are the characteristics of an isosceles triangle?
4. How do you find angles and sides when only one angle is given in a triangle?
5. Can the Law of Sines be used for right-angled triangles?

#### Tip:
Always ensure that the angles and sides match correctly in the Law of Sines, where the angles and their opposite sides are paired up.

Find a and b in an isosceles triangle where the base is 76 cm, the vertex angle is 50°, and the two equal sides are a and b.

Learn how to find the equal sides a and b of an isosceles triangle with a base of 76 cm and a vertex angle of 50°. This solution uses the Law of Sines and the angle sum property of a triangle to solve the problem. Suitable for Grades 7-9.

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