SAT II Math I : Area

Study concepts, example questions & explanations for SAT II Math I

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Example Questions

Example Question #21 : 2 Dimensional Geometry

On the XY plane, line segment AB has endpoints (0, a) and (b, 0). If a square is drawn with segment AB as a side, in terms of a and what is the area of the square?

Possible Answers:

\(\displaystyle A=2(ab)^2\)

Cannot be determined

\(\displaystyle A=b^2+a^2\)

\(\displaystyle A=(a + b) (a - b)\)

\(\displaystyle A=\frac{1}{2} (a + b)\)

Correct answer:

\(\displaystyle A=b^2+a^2\)

Explanation:

Since the question is asking for area of the square with side length AB, recall the formula for the area of a square.

\(\displaystyle A=\text{side}^2\)

Now, use the distance formula to calculate the length of AB.

\(\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

let 

\(\displaystyle \\(x_1,y_1)=(0,a) \\(x_2,y_2)=(b,0)\)

Now substitute the values into the distance formula.

\(\displaystyle \\d=\sqrt{(b-0)^2+(0-2)^2} \\d=\sqrt{b^2+a^2}\)

From here substitute the side length value into the area formula.

\(\displaystyle \\A=\text{side}^2 \\A=\left(\sqrt{b^2+a^2}\right)^2 \\A=b^2+a^2\)

 

Example Question #451 : Sat Subject Test In Math I

Give the area of \(\displaystyle \bigtriangleup ABC\) to the nearest whole square unit, where:

\(\displaystyle AB = 13\)

\(\displaystyle BC = 25\)

\(\displaystyle AC = 17\)

Possible Answers:

\(\displaystyle 55\)

\(\displaystyle 29\)

\(\displaystyle 102\)

Cannot be determined

\(\displaystyle 74\)

Correct answer:

\(\displaystyle 102\)

Explanation:

The area of a triangle, given its three sidelengths, can be calculated using Heron's formula:

\(\displaystyle A = \sqrt{s(s-a)(s-b)(s-c)}\),

where \(\displaystyle a\)\(\displaystyle b\), and \(\displaystyle c\) are the lengths of the sides, and \(\displaystyle s = \frac{a+b+ c}{2}\).

Setting, \(\displaystyle a = AB = 13\)\(\displaystyle b = AC = 17\), and \(\displaystyle c= BC = 25\),

\(\displaystyle s = \frac{13+17+25}{2} = \frac{55}{2} = 27.5\)

and, substituting in Heron's formula:

\(\displaystyle A = \sqrt{27.5(27.5-13)(27.5-17)(27.5-25 )}\)

\(\displaystyle = \sqrt{27.5(14.5)(10.5)(2.5 )}\)

\(\displaystyle = \sqrt{10,467.1875}\)

\(\displaystyle \approx 102\)

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