Question

reference q.14149 calculate, to the nearest tenth, the length of the indicated side in each triangle.
a) a 9.3 4.8
b) 9.6 53° b
c) 7.1 5.6 c
c
c
d) d 38° 16.5
e) 8.2 19° e
e
f) f 33° 20
f
g) 1.9 64° g
g
h) h 50° 2.7
h

Explanation:

Step1: Use Pythagorean theorem for right - angled triangles (a, c)

For right - angled triangle, if the hypotenuse is $c$ and the other two sides are $a$ and $b$, then $c=\sqrt{a^{2}+b^{2}}$ or $a=\sqrt{c^{2}-b^{2}}$. For non - right angled triangles, we use trigonometric ratios like $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

a)

We have a right - angled triangle with hypotenuse $9.3$ and one side $4.8$. Using the Pythagorean theorem $a=\sqrt{9.3^{2}-4.8^{2}}=\sqrt{(9.3 + 4.8)(9.3 - 4.8)}=\sqrt{14.1\times4.5}=\sqrt{63.45}\approx 7.96\approx8.0$.

b)

We know an angle $\theta = 53^{\circ}$ and the adjacent side to the angle is $9.6$. We want to find the opposite side $b$. Using $\tan\theta=\frac{b}{9.6}$, so $b = 9.6\times\tan(53^{\circ})\approx9.6\times1.3270\approx12.7$.

c)

We have a right - angled triangle with sides $5.6$ and $7.1$. Using the Pythagorean theorem, the hypotenuse $c=\sqrt{5.6^{2}+7.1^{2}}=\sqrt{31.36 + 50.41}=\sqrt{81.77}\approx9.0$.

d)

We know an angle $\theta = 38^{\circ}$ and the hypotenuse is $16.5$. We want to find the adjacent side $d$. Using $\cos\theta=\frac{d}{16.5}$, so $d = 16.5\times\cos(38^{\circ})\approx16.5\times0.7880\approx13.0$.

e)

We know an angle $\theta = 19^{\circ}$ and the hypotenuse is $8.2$. We want to find the opposite side $e$. Using $\sin\theta=\frac{e}{8.2}$, so $e = 8.2\times\sin(19^{\circ})\approx8.2\times0.3256\approx2.7$.

f)

We know an angle $\theta = 33^{\circ}$ and the adjacent side to the angle is $20$. We want to find the hypotenuse $f$. Using $\cos\theta=\frac{20}{f}$, so $f=\frac{20}{\cos(33^{\circ})}\approx\frac{20}{0.8387}\approx23.8$.

g)

We know an angle $\theta = 64^{\circ}$ and the adjacent side to the angle is $1.9$. We want to find the opposite side $g$. Using $\tan\theta=\frac{g}{1.9}$, so $g = 1.9\times\tan(64^{\circ})\approx1.9\times2.0503\approx3.9$.

h)

We know an angle $\theta = 50^{\circ}$ and the hypotenuse is $2.7$. We want to find the adjacent side $h$. Using $\cos\theta=\frac{h}{2.7}$, so $h = 2.7\times\cos(50^{\circ})\approx2.7\times0.6428\approx1.7$.

Answer:

a. $8.0$
b. $12.7$
c. $9.0$
d. $13.0$
e. $2.7$
f. $23.8$
g. $3.9$
h. $1.7$