Question

3a. if ab = 24 and ac = 25, find the length of bc.
3b. use the information from 3a to find the perimeter of rectangle abcd.

Explanation:

Step1: Apply Pythagorean theorem

In right - triangle ABC (since ABCD is a rectangle, ∠ABC = 90°), by the Pythagorean theorem \(AC^{2}=AB^{2}+BC^{2}\). We know \(AB = 24\) and \(AC = 25\), so \(BC=\sqrt{AC^{2}-AB^{2}}\).
\[BC=\sqrt{25^{2}-24^{2}}=\sqrt{(25 + 24)(25 - 24)}=\sqrt{49\times1}=\sqrt{49}=7\]

Step2: Calculate the perimeter of the rectangle

The opposite sides of a rectangle are equal. In rectangle ABCD, \(AB = CD = 24\) and \(BC=AD = 7\). The perimeter \(P\) of a rectangle is given by \(P = 2(AB + BC)\).
\[P=2(24 + 7)=2\times31 = 62\]

Answer:

3a. 7
3b. 62