a Draw a triangle with side lengths $$ 10.5 \mathrm{~cm}, 7.5 \mathrm{~cm} $$ and 6.5 cm . b Measure the largest angle of the triangle.

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To draw a triangle with side lengths of $$ 10.5 \, \text{cm} $$, $$ 7.5 \, \text{cm} $$, and $$ 6.5 \, \text{cm} $$, you can follow these steps:

### a. Drawing the Triangle

1. **Draw the longest side**: Start by drawing a line segment that is $$ 10.5 \, \text{cm} $$ long. Label the endpoints as $$ A $$ and $$ B $$.

2. **Construct the second side**: From point $$ A $$, use a compass to measure $$ 7.5 \, \text{cm} $$ and draw an arc above or below the line segment $$ AB $$.

3. **Construct the third side**: From point $$ B $$, measure $$ 6.5 \, \text{cm} $$ with the compass and draw another arc that intersects the first arc.

4. **Label the intersection point**: Label the intersection point of the two arcs as point $$ C $$.

5. **Complete the triangle**: Draw line segments $$ AC $$ and $$ BC $$ to complete the triangle $$ ABC $$.

### b. Measuring the Largest Angle

To find the largest angle of the triangle, you can use the fact that the largest angle is opposite the longest side. In this case, the longest side is $$ AB $$ with a length of $$ 10.5 \, \text{cm} $$.

1. **Identify the largest angle**: The largest angle is $$ \angle C $$, which is opposite side $$ AB $$.

2. **Measure the angle**: Use a protractor to measure $$ \angle C $$ in degrees.

If you do not have a protractor, you can also use the Law of Cosines to calculate the angle:

$$
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
$$

Where:
- $$ a = 7.5 \, \text{cm} $$ (length of side $$ AC $$)
- $$ b = 6.5 \, \text{cm} $$ (length of side $$ BC $$)
- $$ c = 10.5 \, \text{cm} $$ (length of side $$ AB $$)

Substituting the values:

$$
\cos(C) = \frac{(7.5)^2 + (6.5)^2 - (10.5)^2}{2 \cdot 7.5 \cdot 6.5}
$$

Calculating the squares:

$$
\cos(C) = \frac{56.25 + 42.25 - 110.25}{97.5}
$$
$$
\cos(C) = \frac{-11.75}{97.5} \approx -0.1205
$$

Now, find $$ C $$:

$$
C \approx \cos^{-1}(-0.1205) \approx 98.0^\circ
$$

Thus, the largest angle $$ C $$ is approximately $$ 98.0^\circ $$.
To draw the triangle, start with the longest side $$ 10.5 \, \text{cm} $$, then construct the other two sides $$ 7.5 \, \text{cm} $$ and $$ 6.5 \, \text{cm} $$. The largest angle, opposite the longest side, is approximately $$ 98.0^\circ $$.

a Draw a triangle with side lengths \( 10.5 \mathrm{~cm}, 7.5 \mathrm{~cm} \) and 6.5 cm . b Measure the largest angle of the triangle.

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