Question

triangle a is similar to triangle b. the length of the base of triangle b is \\( \frac { 1 } { 2 } \\) times the length of the base of triangle a, and the height of triangle b is \\( \frac { 1 } { 2 } \\) times the height of triangle a. the area of triangle a is 161 square units. what is the area, in square units, of triangle b?

Explanation:

Step1: Recall the formula for the area of a triangle

The area of a triangle is \(A=\frac{1}{2}bh\). Let the base of triangle \(A\) be \(b_A\) and height be \(h_A\), so \(A_A = \frac{1}{2}b_Ah_A=161\). For triangle \(B\), \(b_B=\frac{1}{3}b_A\) and \(h_B=\frac{1}{2}h_A\).

Step2: Calculate the area of triangle \(B\)

The area of triangle \(B\), \(A_B=\frac{1}{2}b_Bh_B\). Substitute \(b_B=\frac{1}{3}b_A\) and \(h_B=\frac{1}{2}h_A\) into the formula: \(A_B=\frac{1}{2}(\frac{1}{3}b_A)(\frac{1}{2}h_A)\). Simplify \(A_B=\frac{1}{12}b_Ah_A\). Since \(A_A=\frac{1}{2}b_Ah_A = 161\), then \(b_Ah_A=322\). Substitute \(b_Ah_A = 322\) into the formula for \(A_B\): \(A_B=\frac{1}{12}\times322=\frac{161}{6}\approx26.83\).

Answer:

\(\frac{161}{6}\)