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555-605 (Medium)|   Geometry|      
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Bunuel
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An isosceles triangle has sides of length x, 2x and 2x. If the area of 16 Nov 2022, 11:37
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69% (01:57) correct 31% (02:44) wrong based on 16 sessions

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An isosceles triangle has sides of length x, 2x and 2x. If the area of the triangle is 25√15, what is the value of x?


A. √15
B. 2√15
C. 10√15
D. 5
E. 10
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E
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An isosceles triangle has sides of length x, 2x and 2x. If the area of 16 Nov 2022, 11:57
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An isosceles triangle has sides of length x, 2x and 2x. If the area of 16 Nov 2022, 12:30
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An isosceles triangle has sides of length x, 2x and 2x. If the area of the triangle is 25√15, what is the value of x?
Solution:-
here the side are x , 2x and 2x
if we put the perpendicular to base then it will bisect the base in 2 equal part
so perpendicular =sqrot(4x^2-x^2/4)
=sqrot(15x^2/4)
so area is 1/2*x*(sqrot(15)x/2)
25sqort15=1/4(sqrot(15)X
x=10
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An isosceles triangle has sides of length x, 2x and 2x. If the area of 16 Nov 2022, 12:56
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In an isosceles triangle, a perpendicular dropped from the common vertex of the two equal sides onto the opposite side bisects the opposite side.

We can drop a perpendicular and find the value of perpendicular in terms of x.

Area of the triangle = 1/2 * base * height

Once the area is found in terms of x, it can be equated with the given area to find the value of x.

The working is shown below.

Option E.
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An isosceles triangle has sides of length x, 2x and 2x. If the area of 16 Nov 2022, 14:28
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Bunuel
An isosceles triangle has sides of length x, 2x and 2x. If the area of the triangle is 25√15, what is the value of x?


A. √15
B. 2√15
C. 10√15
D. 5
E. 10
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Let the length of the height belonging to the base of length x be h. Dropping this height creates two congruent right triangles with hypotenuse 2x, and with legs of lengths x/2 and h. Applying the Pythagorean theorem to one of these triangles, we find:

(2x)² = (x/2)² + h²

4x² = x²/4 + h²

h² = 15x²/4

h = (√15)x/2

Thus, the area of the isosceles triangle, in terms of x, is 1/2 * x * (√15)x/2 = (√15)x²/4. Setting this expression equal to 25√15, we find:

(√15)x²/4 = 25√15

x²/4 = 25

x² = 100

x = ±10

Since x is the length of one side of the isosceles triangle, it cannot be negative. Thus, x must be equal to 10.

Answer: E
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