Question

solve each triangle. round answers to the nearest tenth.

Explanation:

Step1: Use Pythagorean theorem for right - angled triangles

For a right - angled triangle with sides \(a\), \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\) (when \(c\) is the hypotenuse). For non - right angled triangles, we can use trigonometric ratios \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) and the fact that the sum of angles in a triangle is \(180^{\circ}\).

Triangle 1:

Let's assume the right - angled triangle with legs \(a = 9\) and \(b = 12\).

Step2: Find the hypotenuse

Using the Pythagorean theorem \(c=\sqrt{9^{2}+12^{2}}=\sqrt{81 + 144}=\sqrt{225}=15\).
To find the angles, \(\tan A=\frac{12}{9}=\frac{4}{3}\), so \(A=\arctan(\frac{4}{3})\approx53.1^{\circ}\), and \(B = 90^{\circ}-A\approx36.9^{\circ}\)

Triangle 2:

We know one angle \(A = 38^{\circ}\) and one side \(a = 12.5\).

Step2: Find angle \(B\)

Since it's a right - angled triangle, \(B=90^{\circ}-38^{\circ}=52^{\circ}\)
To find the other sides, if the side opposite to \(A\) is \(a = 12.5\), and \(\sin A=\frac{a}{c}\), then \(c=\frac{12.5}{\sin38^{\circ}}\approx\frac{12.5}{0.616}\approx20.3\). And \(b=\sqrt{c^{2}-a^{2}}=\sqrt{20.3^{2}-12.5^{2}}=\sqrt{(20.3 + 12.5)(20.3-12.5)}=\sqrt{32.8\times7.8}\approx16.0\)

Triangle 3:

Let the sides of the right - angled triangle be \(a = 2\) and \(c = 3\) (where \(c\) is the hypotenuse).

Step2: Find side \(b\)

Using the Pythagorean theorem \(b=\sqrt{c^{2}-a^{2}}=\sqrt{9 - 4}=\sqrt{5}\approx2.2\)
\(\sin A=\frac{2}{3}\), so \(A=\arcsin(\frac{2}{3})\approx41.8^{\circ}\), and \(B=90^{\circ}-A\approx48.2^{\circ}\)

Triangle 4:

We know one angle \(A = 59^{\circ}\) and hypotenuse \(c = 16\)

Step2: Find side \(a\)

\(a = c\times\sin A=16\times\sin59^{\circ}\approx16\times0.857\approx13.7\)
\(b = c\times\cos A=16\times\cos59^{\circ}\approx16\times0.515\approx8.2\)
\(B=90^{\circ}-59^{\circ}=31^{\circ}\)

Answer:

For triangle 1: Sides are 9, 12, 15; Angles are approximately \(36.9^{\circ}\), \(53.1^{\circ}\), \(90^{\circ}\)
For triangle 2: Sides are approximately 12.5, 16.0, 20.3; Angles are \(38^{\circ}\), \(52^{\circ}\), \(90^{\circ}\)
For triangle 3: Sides are approximately 2, 2.2, 3; Angles are approximately \(41.8^{\circ}\), \(48.2^{\circ}\), \(90^{\circ}\)
For triangle 4: Sides are approximately 8.2, 13.7, 16; Angles are \(31^{\circ}\), \(59^{\circ}\), \(90^{\circ}\)