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17 May 2018, 05:06
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B
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If each of the three triangles has an area of 18, what is the perimeter of the figure they form?
A) 54
B) 36 + 18√2
C) 48 + 18√2
D) 54√2
E) 54 + 27√2
Attachment:
Untitled.png [ 5.45 KiB | Viewed 1926 times ]
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17 May 2018, 05:29
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Bunuel
.
Not very comfortable with drawing here.
So I am representing T1 - Upper Triangle. T2 - Lower Triangle. T3- bottom most triangle.
Line l1 perpendiualr to L3. Right angle triangle. = 90'
L1 parallel to L2. Alternate Interior Angles are equal. 45'.
Third angle = 180-90-45 = 45'
It is Isosceles Right Triangle. base = height .
area of Triangle = 1/2 * base * height = 18
1/2 * base * height = 18
base * height = 18*2 =36.
base = height . It implies . base * base = 36.
base = height = 6
Hypotenuse = \sqrt{\(6^2 + 6^2\)} = 6sqrt(2)
Perimeter of 1 triangle = 12 + 6*sqrt(2)
You can repeat the same for both the triangle.All will have ame area. Hence same dimension of sides.
Perimeter of the figure = 3 * perimeter of 1 triangle = 3 (12 + 6*sqrt(2)) = 36 + 18 * sqrt(2).
Ans - B
17 May 2018, 18:36
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Bunuel
Attachment:
perimeter.png [ 33.42 KiB | Viewed 1725 times ]
Find other angle measures; these figures are probably special triangles
Perimeter: Find side length from area, hypotenuse length from special properties
• 90° angles at A and E (two angles at E)
Line 1 || Line 2
Line 1 ⍊ Line 3, thus
Line 2 ⍊ Line 3
a line perpendicular to one of two parallel lines
is perpendicular to the other line also
• ∠ D = 45 = ∠ B
(alternate interior angles of parallel lines 1 and 2 cut by transversal)
• ∠ B = 45° = ∠ F
(alternate interior angles of parallel lines 4 and 5 cut by transversal)
• ∠ G = ∠ C (two angles at C) = 45°
Each triangle so far has one 90° and one 45° angle. The other is 45°
(90 + 45) = 135° and then (180°-135°) = 45°
The three triangles are right isosceles triangles
Sides opposite equal angles are equal.
• Perimeter? Need side lengths. Derive them from area
Area of right isosceles triangle, \(A=\frac{s^2}{2}\)
Same as \(\frac{b*h}{2}\); here \(b = h\)
\(\frac{s^2}{2} = 18\)
\(s^2 = 36\)
\(s = 6\)
Hypotenuse?*
Isosceles right triangles have sides in ratio \(x: x: x\sqrt{2}\)
(\(x\) and \(x\) are opposite 45° angles, \(x\sqrt{2}\) is opposite 90° angle)
\(s = 6\) corresponds with \(x\)
Hypotenuse corresponds with \(x\sqrt{2} = 6\sqrt{2}\)
Perimeter:
Legs: \(6 * 6 = 36\)
Hypotenuses: \(3 * 6\sqrt{2} = 18\sqrt{2}\)
\(P = 36 + 18\sqrt{2}\)
Answer B
*Or use Pythagorean theorem: \(Side^2+Side^2=Hypotenuse^2\)
\(8^2+8^2=H^2\)
\(H^2=(64+64)=128\)
\(\sqrt{H^2}=\sqrt{128}\)
\(\sqrt{H^2}=\sqrt{64*2}=(\sqrt{64}*\sqrt{2})\)
\(H=8\sqrt{2}\)
