Question : In $\triangle \mathrm{PQR}, \angle \mathrm{Q}=90$°, and QX is perpendicular to PR. If PQ = 15 cm and PR = 25 cm, what is the PX : XR ratio?
Option 1: 2 : 3
Option 2: 7 : 18
Option 3: 1 : 4
Option 4: 9 : 16
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Correct Answer: 9 : 16
Solution :
If a perpendicular is drawn from the vertex of the right angle of a right-angled triangle to the hypotenuse, then it divides the hypotenuse into two segments.
The lengths of these segments are proportional to the lengths of the other two sides.
This is based on the principle of similar triangles.
Using the principle of similar triangles, since $\triangle$PQR ~ $\triangle$PQX, we have:
$\frac{PX}{PQ}$ = $\frac{PQ}{PR}$
Plugging in the given values:
$\frac{PX}{15}$ = $\frac{15}{25}$
$\therefore$ PX = 9 cm
So, XR = PR – PX = 25 – 9 = 16
Therefore, the ratio of PX to XR is 9 : 16.
hence, the correct answer is 9 : 16.
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