Geometry Smart Math

[Smart Math] Geometry Problem 13

Here’s and example of a SMART MATH problem for GEOMETRY.



The equal sides of an isosceles triangle are even integers only. What is the area of the triangle if the perimeter is 7 units?

  1. 7 sq. units
  2. 3 sq. units
  3. 8 sq. units
  4. \frac{3\sqrt{7}}{4}sq. units
  5. 3\sqrt{7}sq. units

The Usual Method

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Even integers are 2, 4, 6 … Now to have the perimeter as 7 i.e. a + a + b = 7, where ‘1’ is the length of equal side and is an even integer.

Hence, 2a + b = 7

Now a can take the value of 2 only.

So 2(2) + b = 7

\therefore b = 3

Hence dimension of the triangle = 2, 2, and 3 units.

Hence, area of this triangle using semi-perimeter formula (Area = \sqrt{s(s-a)(s-b)(s-c)}, where s=\frac{a+b+c}{2} = semi-perimeter)
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In this case, s = \frac{2+2+3}{2}=\frac{7}{2}
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\therefore Area = \sqrt{\frac{7}{2}\left( \frac{7}{2}-2 \right)\left( \frac{7}{2}-2 \right)\left( \frac{7}{2}-3 \right)}
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= \sqrt{\frac{7}{2}\left( \frac{3}{2} \right)\left( \frac{3}{2} \right)\left( \frac{1}{2} \right)}
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= \sqrt{\frac{63}{16}}=\sqrt{\frac{7}{4}\times \frac{9}{4}}=\frac{3\sqrt{7}}{4}sq. units

(Ans: 4)

Estimated Time to arrive at the answer = 45 seconds.

Using Technique

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Once you reach to a the level of identifying the dimensions of the triangle (2, 2 and 3), put the values as \sqrt{\frac{7}{2}\left( \frac{7}{2}-2 \right)\left( \frac{7}{2}-2 \right)\left( \frac{7}{2}-3 \right)}, to get the answer.

Now, do not solve this. Just by observing it, you will see that there are four 2s in the denominator with the \sqrt{{}} sign. Hence, the denominator will be \sqrt{2^{4}} = 4. Also 7 is a prime number so \sqrt{7}\times … will have \sqrt{7}(or one of its form) as the numerator. Hence the answer should have \frac{n\sqrt{7}}{4}. This format is only in option ‘4’.

(Ans: 4)

Estimated Time to arrive at the answer = 15 seconds.
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