In Trigonometry, SAS corresponds to a facet-attitude-side triangle.
What in case you consider how to locate the angle of a triangle given 2 aspects and 1 perspective. allow us to let you know!
Suppose we must solve triangle SAS given as under:
We are given:
\(a = 5\)
\(b = 7\)
\(γ = 45^{\text{o}}\)
Third Side c:
By using the law of cosines:
\(c = \sqrt{\left(5\right)^{2} + \left(7\right)^{2} - 2 \cdot 5 \cdot 7 \cdot \cos\left(45^{\text{o}}\right)}\)
\(c = \sqrt{25 + 49 - 70 \cdot 0.707}\)
\(c = \sqrt{25 + 49 - 49.49}\)
\(c = \sqrt{24.51}\)
\(c = 4.95\)
Perimeter:
\(p = a + b + c\)
\(p = 5 + 7 + 4.95\)
\(p = 16.95\)
Semiperimeter:
\(s = \dfrac{p}{2}\)
\(s = \dfrac{16.95}{2}\)
\(s = 8.475\)
Area:
By using Heron’s formula: \(A = \sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\)
\(A = \sqrt{8.475 \cdot \left(8.475 - 5\right) \cdot \left(8.475 - 7\right) \cdot \left(8.475 - 4.95\right)}\)
\(A = \sqrt{8.475 \cdot 3.475 \cdot 1.475 \cdot 3.525}\)
\(A = \sqrt{153.3}\)
\(A = 12.38\)
Height of Triangle:
\(h_{a} = \dfrac{2A}{a}\)
\(h_{a} = \dfrac{2 \cdot 12.38}{5}\)
\(h_{a} = 4.952\)
\(h_{b} = \dfrac{2A}{b}\)
\(h_{b} = \dfrac{2 \cdot 12.38}{7}\)
\(h_{b} = 3.537\)
\(h_{c} = \dfrac{2A}{c}\)
\(h_{c} = \dfrac{2 \cdot 12.38}{4.95}\)
\(h_{c} = 5.004\)
Inner Angles:
By using the law of sines:
\(\dfrac{b}{\sinꞵ} = \dfrac{c}{\sin\gamma}\)
\(sinꞵ = \dfrac{b}{c} \cdot \sin\gamma\)
\(sinꞵ = \dfrac{7}{4.95} \cdot \sin\left(45^{\text{o}}\right)\)
\(sinꞵ = 1.414 \cdot 0.707\)
\(sinꞵ = 1\)
\(ꞵ = 90^{\text{o}}\)
Now using the supplementary angle measurement:
\(\alpha + ꞵ + \gamma = 180^{\text{o}}\)
\(\alpha + 90^{\text{o}} + 45^{\text{o}} = 180^{\text{o}}\)
\(\alpha = 45^{\text{o}}\)
Inradius:
\(r = \dfrac{A}{s}\)
\(r = \dfrac{12.38}{8.475}\)
\(r = 1.46\)
Circumradius:
\(R = \dfrac{a \cdot b \cdot c}{4 \cdot r \cdot s}\)
\(R = \dfrac{5 \cdot 7 \cdot 4.95}{4 \cdot 1.46 \cdot 8.475}\)
\(R = 7.03\)
Medians:
\(m_{a} = \sqrt{\dfrac{2b^2 + 2c^2 - a^2}{2}}\)
\(m_{a} = \sqrt{\dfrac{2 \cdot 7^2 + 2 \cdot 4.95^2 - 5^2}{2}}\)
\(m_{a} = 6.85\)
\(m_{b} = \sqrt{\dfrac{2c^2 + 2a^2 - b^2}{2}}\)
\(m_{b} = \sqrt{\dfrac{2 \cdot 4.95^2 + 2 \cdot 5^2 - 7^2}{2}}\)
\(m_{b} = 5.12\)
\(m_{c} = \sqrt{\dfrac{2a^2 + 2b^2 - c^2}{2}}\)
\(m_{c} = \sqrt{\dfrac{2 \cdot 5^2 + 2 \cdot 7^2 - 4.95^2}{2}}\)
\(m_{c} = 6.98\)
If you want to use our SAS calculator, examine on and apprehend the following guide!
Input:
Output:
