In isosceles triangle APS, angle P is 54°, and AP = AS. Which is larger, AP or PS?

Question

08-04-2025
Mathematics

High School

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In isosceles triangle APS, angle P is 54°, and AP = AS. Which is larger, AP or PS?

Answer :

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sujatavse27 sujatavse27 · Answer 1 · 2025-04-20T02:51:11-04:00

Final answer:

In the isosceles triangle APS, with angles at A and S being 63° each and angle P being 54°, AP (equal to AS) is larger than PS, as sides are proportional to the sines of their opposite angles and sine of 63° is greater than sine of 54°.

Explanation:

Given angle P = 54° and AP = AS. In an isosceles triangle, the angles opposite the equal sides are also equal. Since angle P is 54°, the angles at A and S combined must sum to 180° - 54° = 126°. Because these are the angles at the base of the isosceles triangle and the triangle is isosceles (AP = AS), these angles are equal, making each angle 63° (126° divided by 2).

Using the law of sines, we can infer that in a triangle, sides are proportional to the sines of their opposite angles. Since the sine of 63° (the angles at A and S) is larger than the sine of 54° (the angle at P), it indicates that AP (or AS) is longer than PS. Therefore, AP is larger than PS in the isosceles triangle APS, where angle P = 54° and AP = AS.