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An acute-angled isosceles triangle has two of its sides

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Director
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An acute-angled isosceles triangle has two of its sides  [#permalink]

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New post 02 Aug 2019, 06:31
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A
B
C
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E

Difficulty:

  65% (hard)

Question Stats:

42% (02:02) correct 58% (02:21) wrong based on 31 sessions

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An acute-angled isosceles triangle has two of its sides equal to 14 and 25. Find the area of this triangle.

A. 175
B. 50√6
C. 168
D. 140
E. None of these

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An acute-angled isosceles triangle has two of its sides  [#permalink]

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New post 02 Aug 2019, 06:57
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If it is acute-angled isosceles triangle, sides will be 25(side),25(side) and 14 (base)
—> As we know, the altitude of isosceles triangle bisect the base—>
\((Altitude)^2\)=\(25^2\) —\(7^2\)=625–49=576
—> Altitude =24
—> the area of triangle is equal to
\(\frac{(Altitude*base)}{2}\)= \(\frac{24*14}{2}\)= 168

The answer choice is C

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Re: An acute-angled isosceles triangle has two of its sides  [#permalink]

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New post 09 Aug 2019, 03:34
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If a, b and c are the sides of an acute angled triangle, with c being the longest side, then
\(a^2 + b^2 > c^2\).

Conisdering this information, it’s easy to figure out that the equal sides cannot be 14. If the equal sides are 14, then the biggest side becomes 25. But \(14^2 + 14^2\) is not greater than \(25^2\).

So, the equal sides have to be 25. If the equal sides are 25, the unequal side is 14.

In an isosceles triangle, the height dropped from the vertex containing the unequal angle also acts as the perpendicular bisector of the unequal side.
Here’s a figure showing the same:

Attachment:
9th Aug - Reply 3.JPG
9th Aug - Reply 3.JPG [ 14.42 KiB | Viewed 205 times ]


So, the original triangle can now be divided into two equal right angled triangles, of base 7 and hypotenuse 25. This means that the height of the triangle should be 24.
The area of each of these right angled triangles = ½ * 7 * 24 = 84. Therefore, the area of the bigger triangle = 84 *2 = 168 sq. units.

The correct answer option is C.
Hope this helps!
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Re: An acute-angled isosceles triangle has two of its sides   [#permalink] 09 Aug 2019, 03:34
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