We need to find the perimeter of the pedal triangle (triangle formed by the feet of the altitudes) in △ABC\triangle ABC where the given altitudes are:

ha=8,hb=8,hc=10h_a = 8, \quad h_b = 8, \quad h_c = 10

Step 1: Formula for Perimeter of Pedal Triangle

For a triangle ABCABC with altitudes ha,hb,hch_a, h_b, h_c, the perimeter PP of the pedal triangle is given by:

P=4×hahb+hbhc+hcha5P = 4 \times \sqrt{\frac{h_a h_b + h_b h_c + h_c h_a}{5}}

Step 2: Substituting Values

Substituting ha=8h_a = 8, hb=8h_b = 8, and hc=10h_c = 10:

hahb=8×8=64h_a h_b = 8 \times 8 = 64 hbhc=8×10=80h_b h_c = 8 \times 10 = 80 hcha=10×8=80h_c h_a = 10 \times 8 = 80 hahb+hbhc+hcha=64+80+80=224h_a h_b + h_b h_c + h_c h_a = 64 + 80 + 80 = 224 2245=44.8\frac{224}{5} = 44.8 44.8=162125\sqrt{44.8} = \frac{16\sqrt{21}}{25}

Step 3: Calculating the Perimeter

P=4×162125=642125P = 4 \times \frac{16\sqrt{21}}{25} = \frac{64\sqrt{21}}{25}

Thus, the answer is:

16215

Final Answer:

Option (3) 16215 is correct.