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13 Feb 2019, 08:04
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In the figure above, AC is a diameter of the circle. What is the area of triangle ABC?
(1) The area of the circle is \(4900\pi\) and the length of BC is 84.
(2) The perimeter of triangle ABC is 336 and the length of AB is \(\frac{4}{5}\) times the length of BC.
(1) The area of the circle is \(4900\pi\) and the length of BC is 84.
(2) The perimeter of triangle ABC is 336 and the length of AB is \(\frac{4}{5}\) times the length of BC.
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13 Feb 2019, 08:22
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PeepalTree
Since AC is the diameter of the circle, angle B = 90
(1) The area of the circle is \(4900\pi\) and the length of BC is 84.
Area of the circle will give the diameter AC, and another side BC is known. Now, since it is a right angled triangle, we can find the third side and thus the area of triangle can be found.
Sufficient
(2) The perimeter of triangle ABC is 336 and the length of AB is \(\frac{4}{5}\) times the length of BC.
The three sides are x, (4/5)x and \(\sqrt{x^2+(4/5)x^2}\) and sum of these three sides is 336
x+ (4/5)x + \(\sqrt{x^2+(4/5)x^2}\) =336. we can find the value of x and thus the area.
Sufficient
D
You don't require to find the area and above would be sufficient without solving but let us solve it incase we find this in a PS question.
(1) The area of the circle is \(4900\pi\) and the length of BC is 84.
Area = \(\pi*r^2=4900\pi....r=70\), so AC =2*70=140 as it is the diameter.
Thus AB = \(\sqrt{AC^2-BC^2}=\sqrt{140^2-84^2}=\sqrt{28^2(5^2-3^2)}=28*4=112\)
Area = \(\frac{1}{2}*84*112=4704\)
(2) The perimeter of triangle ABC is 336 and the length of AB is \(\frac{4}{5}\) times the length of BC.
The three sides are x, (4/5)x and \(\sqrt{x^2+(4/5)x^2}\) and sum of these three sides is 336
x+ (4/5)x + \(\sqrt{x^2+(4/5)x^2}\) =336. \(x(1+(4/5)+(3/\sqrt{5})=336\)
You can find answer but the statement I and II do not match
KanishkM
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13 Feb 2019, 08:38
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PeepalTree
Both statements are sufficient, D
1) Area of circle = \(4900\pi\) , length of BC = 84
Since AC is a diameter, < ABC = 90, From given area, i can calculate radius, from which the length of AC
I can calculate AB through Pythagoras theorem.
Area of Triangle can be calculated.
2) AB + BC + CA = 336----------(a) & AB = 4/5 * BC
AC = root of AB^2 + BC^2 -------(b)
I can substitute (b) in (a), and make an equation in only one variable that is BC
Once we get BC, we can get AB. We know both the lengths of AB & BC
Area of Triangle can be calculated.
