Triangle in a Semicircle

Geometry Level 2

P P is a point on a semicircle with diameter A B ‾ = 7 \overline{AB}=7 . What is the maximum value of 3 A P ‾ + 4 B P ‾ ? 3\overline{AP}+4\overline{BP}?

34 36 33 35

View wiki
Mahindra Jain by Mahindra Jain

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

4 solutions

Renato Javier
Mar 14, 2014

Point P always makes a right triangle when joined with the edges of a diameter. Let AP=x and BP =y. By Pythagorean Theorem, x^2 + y^2 = 49. Therefore x= sqrt of (49-y^2). Let S = 3x + 4y . Substitute the value of x in terms of y, then differentiate with respect to y. Equate to zero and simplify to get y = 28/5. Substitute this value of y to get x = 21/5. Substitute x and y to get S = 35.

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Just observe that A P B = 90 ° APB=90° and then use trignometry to sovle this problem.

mietantei conan - 7 years ago

from properties of triangles bCos C+ cCos B = a . it can be proved

Dharma Teja - 7 years ago
Kirsten Gouria
Feb 26, 2014

use derivative test for maxima of function

3 Helpful 0 Interesting 0 Brilliant 0 Confused

yup.

Renato Javier - 7 years, 3 months ago
Dean Clidoro
Mar 13, 2014

Solution:

  1. Put point P at the mid of the circumference

  2. This creates a n isosceles triangle with angle 45

    a. so one side is (7/2)/cos45 = 4.95

  3. Try making the triangle into a 30 - 60 - 90 so it will be maximum

    a. long side = 7cos 30 = 6 ; short side is 7sin30 = 3.5

    b. the sum of side is greater and maximum compared to 45 deg triangle

  4. sum = 3(3.5) + 4(6) = 35......check

2 Helpful 0 Interesting 0 Brilliant 0 Confused

tsumamba lang yan pre.

Renato Javier - 7 years, 3 months ago

Solution: Lets call AP=a, BP=b and the angle PBA alpha then sin(alpha)=a/7 and cos(alpha)=b/7 because P always makes a right triangle then that means 3a+4b=21sin(alpha)+28cos(alpha). And is known that the maximum of that expression is sqrt(21^2+28^2)=35

Bruno Tenorio - 7 years, 1 month ago

Thanks for the good solution.

rugved dhore - 7 years ago
Oscar Carreón
Apr 14, 2015

For the maximum value of AP or BP, we set P= the radius from the point in the middle of AB and the semicircle, call it D [ D|D=AB/2 ].

Now, DP=3.5 and we have that angles PDA = PDB =90º. We know that AD = BD =3.5, which is equal to DP. By the Pythagorean Theorem, therefore, we have: (AD^2 + DP^2) = AP^2 –> AP^2 = (3.5^2 + 3.5^2).

Just apply square root to AP^2. Remember AP = BP so:

3AP + 4BP = 34.64823228

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...