Question

one of the altitudes of the parallelogram shown is \\(\sqrt{225}\\) units which corresponds to side ab. what is the length of the other altitude?\
\\(\bigcirc\\) 5 units\
\\(\bigcirc\\) 6 units\
\\(\bigcirc\\) 30 units\
\\(\bigcirc\\) \\(\sqrt{40}\\) units

Explanation:

Step1: Find length of AB

First, identify coordinates: A(1,4), B(7,2). Use distance formula:
$$AB = \sqrt{(7-1)^2 + (2-4)^2} = \sqrt{6^2 + (-2)^2} = \sqrt{36+4} = \sqrt{40}$$

Step2: Calculate area of parallelogram

Area = base × height. Use AB as base, given altitude $\sqrt{22.5}$:
$$Area = \sqrt{40} \times \sqrt{22.5} = \sqrt{40 \times 22.5} = \sqrt{900} = 30$$

Step3: Find length of BC

Coordinates: B(7,2), C(10,6). Use distance formula:
$$BC = \sqrt{(10-7)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$$

Step4: Solve for the other altitude

Let $h$ = other altitude (corresponding to BC). Area = BC × $h$:
$$30 = 5 \times h \implies h = \frac{30}{5} = 6$$

Answer:

6 units