In right-angled triangle PTS, angle S = 90 °. PT = 17 cm. Find legs PS and TS.

By the Pythagorean theorem:
PT ^ 2 = PS ^ 2 + TS ^ 2;
PS ^ 2 + TS ^ 2 = 17 ^ 2;
PS ^ 2 + TS ^ 2 = 289.
It is known that the tangent of the angle P is 0.25. Tangent is the ratio of the opposite leg to the adjacent one, then:
TS / PS = 25/100;
TS / PS = 1/4.
We get a system of equations with two unknowns:
PS ^ 2 + TS ^ 2 = 289 – equation 1;
TS / PS = 1/4 – Equation 2.
In the second equation, in proportion, we express TS through PS:
TS = 1 * PS / 4;
TS = PS / 4.
Substitute the resulting expression into the first equation:
PS ^ 2 + (PS / 4) ^ 2 = 289.
Let’s solve the resulting equation with one unknown:
PS ^ 2 + PS ^ 2/16 = 289;
(16PS ^ 2 + PS ^ 2) / 16 = 289;
17PS ^ 2/16 = 289;
PS ^ 2 = 289 * 16/17;
PS ^ 2 = 17 * 16;
PS = √17 * 16;
PS = 4√17 cm.
Substitute the resulting PS value into the TS expression:
TS = PS / 4;
TS = 4√17 / 4 = √17 (cm).
Answer: PS = 4√17 cm, TS = √17 cm.