Question

lesson 16: finding all the unknown values in triangles
cool down: calculate and check
in the diagram, triangle abc is similar to triangle abc. calculate the length of side ab. then check your answer using a different method.
$a^2 + b^2 = c^2$
$ac = 4$
$bc = 3$
$ac^2 + bc^2 = ab^2$
$16 + 9 = ab^2$
$25 = ab^2$
$ab = \sqrt{25} = 5$

Explanation:

Step1: Use Pythagorean theorem

For right triangle \(ABC\), \(AC = 4\), \(BC = 3\). By Pythagorean theorem \(AC^{2}+BC^{2}=AB^{2}\). Substitute values: \(4^{2}+3^{2}=AB^{2}\), so \(16 + 9=AB^{2}\), \(25 = AB^{2}\).

Step2: Solve for \(AB\)

Take square root: \(AB=\sqrt{25}=5\).

Step3: Check with similarity

Triangles \(ABC\) and \(A'B'C'\) are similar. Ratio of \(AC\) to \(A'C'\) is \(\frac{4}{6}=\frac{2}{3}\). Ratio of \(AB\) to \(A'B'\) should be same. \(A'B' = 7.5\), check \(\frac{AB}{7.5}=\frac{2}{3}\), so \(AB=\frac{2}{3}\times7.5 = 5\), matches.

Answer:

The length of side \(AB\) is \(5\).