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09 Jan 2025, 00:43
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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Time
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Quantity A
The perimeter of triangle ABC
Quantity B
5
Attachment:
4.jpg [ 13.28 KiB | Viewed 199 times ]
11 Jan 2025, 02:39
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OE
Although there seems to be very little information here, the two small triangles that comprise triangle ABC may seem familiar. First, fill in the additional angles in the diagram:
With the additional angles filled in, it is clear that the two smaller triangles are special right triangles: a 45–45–90 triangle and a 30– 60–90 triangle. You know the ratios of the side lengths for each of these triangles. For a 45–45–90 triangle, the ratio is \(x : x : x \sqrt{2}\).
In this diagram, the value of x is 1 (side BD), so AD is 1 and AB is \(\sqrt{2}\)
For a 30–60–90 triangle, the ratio is \(x : x[m]3\) : 2x[/m]. In this diagram, x is 1 (side BD), so DC is \(\sqrt{3}\) and BC is 2:
Now calculate the perimeter of triangle ABC:
Quantity A
\(1+2+\sqrt{2}+\sqrt{3}\)
Quantity B
5
Now you need to compare this sum to 5. A good approximation of \(\sqrt{2}\) is 1.4 and a good approximation of is \(\sqrt{3}\) 1.7:
In fact, simply knowing that each square root is greater than 1 would let you conclude that, Quantity A is greater.
Alternatively, you could use the calculator to compute Quantity A.
D3.jpg [ 33.46 KiB | Viewed 177 times ]
D2.jpg [ 32.09 KiB | Viewed 174 times ]
D1.jpg [ 22.34 KiB | Viewed 175 times ]
Although there seems to be very little information here, the two small triangles that comprise triangle ABC may seem familiar. First, fill in the additional angles in the diagram:
With the additional angles filled in, it is clear that the two smaller triangles are special right triangles: a 45–45–90 triangle and a 30– 60–90 triangle. You know the ratios of the side lengths for each of these triangles. For a 45–45–90 triangle, the ratio is \(x : x : x \sqrt{2}\).
In this diagram, the value of x is 1 (side BD), so AD is 1 and AB is \(\sqrt{2}\)
For a 30–60–90 triangle, the ratio is \(x : x[m]3\) : 2x[/m]. In this diagram, x is 1 (side BD), so DC is \(\sqrt{3}\) and BC is 2:
Now calculate the perimeter of triangle ABC:
Quantity A
\(1+2+\sqrt{2}+\sqrt{3}\)
Quantity B
5
Now you need to compare this sum to 5. A good approximation of \(\sqrt{2}\) is 1.4 and a good approximation of is \(\sqrt{3}\) 1.7:
In fact, simply knowing that each square root is greater than 1 would let you conclude that, Quantity A is greater.
Alternatively, you could use the calculator to compute Quantity A.
Attachment:
D3.jpg [ 33.46 KiB | Viewed 177 times ]
Attachment:
D2.jpg [ 32.09 KiB | Viewed 174 times ]
Attachment:
D1.jpg [ 22.34 KiB | Viewed 175 times ]
