Question

lesson 9: side length quotients in similar triangles
cool down: similar sides
the 2 triangles shown are similar.
find the side proportions of a and b for the larger triangle.

Explanation:

Step1: Set up proportion for side \(a\)

Since the triangles are similar, the ratios of corresponding sides are equal. For the vertical sides: \(\frac{1.4}{a}=\frac{2.1}{2.1 + b}\). Wait, no, better use the ratio of the two vertical - horizontal side pairs. The ratio of the vertical sides to the horizontal sides in similar triangles is the same. So \(\frac{1.4}{2.1}=\frac{2.1}{a}\).
Cross - multiply: \(1.4a = 2.1\times2.1\).

Step2: Solve for \(a\)

\(a=\frac{2.1\times2.1}{1.4}=\frac{4.41}{1.4}=3.15\).

Step3: Set up proportion for side \(b\)

Use the ratio of the hypotenuse - like sides (the non - right - angle sides). Let the length of the hypotenuse of the small triangle be \(c_1=\sqrt{1.4^{2}+2.1^{2}}=\sqrt{1.96 + 4.41}=\sqrt{6.37}\), and for the large triangle \(c_2=\sqrt{(1.4 + b)^{2}+a^{2}}\). But an easier way is to use the ratio of the vertical sides. The ratio of the vertical sides of the two similar triangles is \(\frac{1.4}{2.1}\). The ratio of the non - vertical, non - horizontal sides (the ones along the straight line) is also \(\frac{1.4}{2.1}\). Let the length of the non - vertical, non - horizontal side of the small triangle be \(x = 1.4\) (assuming it's a right - triangle - like proportion setup, and the length of the non - vertical, non - horizontal side of the large triangle is \(x + b\). So \(\frac{1.4}{1.4 + b}=\frac{2.1}{a}\). Substitute \(a = 3.15\). \(\frac{1.4}{1.4 + b}=\frac{2.1}{3.15}\). Cross - multiply: \(2.1\times(1.4 + b)=1.4\times3.15\). \(2.94+2.1b = 4.41\). \(2.1b=4.41 - 2.94\). \(2.1b = 1.47\). \(b=\frac{1.47}{2.1}=0.7\).

Answer:

\(a = 3.15\), \(b = 0.7\)