Question

9)
triangle xyz with xy = 12.5 cm, yz = 13 cm, xz = 16 cm

Explanation:

Step1: Identify the triangle sides

We have triangle \( XYZ \) with sides \( XY = 12.5 \, \text{cm} \), \( YZ = 13 \, \text{cm} \), and \( XZ = 16 \, \text{cm} \). Let's assume we want to find the area using Heron's formula. First, find the semi - perimeter \( s=\frac{a + b + c}{2}\), where \( a = 12.5 \), \( b = 13 \), \( c = 16 \).
\( s=\frac{12.5+13 + 16}{2}=\frac{41.5}{2}=20.75 \, \text{cm} \)

Step2: Apply Heron's formula

Heron's formula for the area of a triangle is \( A=\sqrt{s(s - a)(s - b)(s - c)} \)
Substitute the values:
\( A=\sqrt{20.75(20.75 - 12.5)(20.75 - 13)(20.75 - 16)} \)
\(=\sqrt{20.75\times8.25\times7.75\times4.75} \)
First, calculate \( 20.75\times8.25 = 20.75\times(8+\frac{1}{4})=20.75\times8+20.75\times\frac{1}{4}=166 + 5.1875 = 171.1875 \)
\( 7.75\times4.75=(8 - 0.25)(4 + 0.75)=8\times4+8\times0.75-0.25\times4 - 0.25\times0.75=32 + 6-1 - 0.1875 = 36.8125 \)
Then, \( 171.1875\times36.8125\approx171.1875\times36.8125 \)
\( 171.1875\times36.8125=(170 + 1.1875)(36+0.8125)=170\times36+170\times0.8125+1.1875\times36+1.1875\times0.8125 \)
\( = 6120+138.125 + 42.75+0.96484375=6120 + 138.125=6258.125+42.75 = 6300.875+0.96484375 = 6301.83984375 \)
\( A=\sqrt{6301.83984375}\approx79.38 \, \text{cm}^2 \)

(If we assume another approach, like using the formula for the area of a triangle with base \( XZ = 16 \) and finding the height \( h \) from \( Y \) to \( XZ \). Let the height be \( h \), and let the base be divided into two segments \( m \) and \( n \) such that \( m + n=16 \), and by Pythagoras: \( h^{2}+m^{2}=12.5^{2} \) and \( h^{2}+n^{2}=13^{2} \). Subtract the first equation from the second: \( n^{2}-m^{2}=13^{2}-12.5^{2}=(13 - 12.5)(13 + 12.5)=0.5\times25.5 = 12.75 \). Also, \( n^{2}-m^{2}=(n - m)(n + m) \), since \( n + m = 16 \), then \( 16(n - m)=12.75\Rightarrow n - m=\frac{12.75}{16}=0.796875 \). Solve the system \( n + m = 16 \) and \( n - m = 0.796875 \). Adding the two equations: \( 2n=16.796875\Rightarrow n = 8.3984375 \), \( m = 16 - 8.3984375 = 7.6015625 \). Then \( h^{2}=12.5^{2}-m^{2}=156.25-(7.6015625)^{2}=156.25 - 57.783203125 = 98.466796875 \), \( h=\sqrt{98.466796875}\approx9.923 \). Then area \( A=\frac{1}{2}\times16\times9.923\approx79.38 \, \text{cm}^2 \))

Answer:

The area of triangle \( XYZ \) is approximately \( 79.4 \, \text{cm}^2 \) (rounded to one decimal place)