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  • Two sides of a triangle are 5 cm and 12 cm long. The measure of the third side is an integer in cm. If the triangle is an obtuse triangle, then how many such triangles ar

Two sides of a triangle are 5 cm and 12 cm long. The measure of the third side is an integer in cm. If the triangle is an obtuse triangle, then how many such triangles are possible?

Option: 1

6


Option: 2

7


Option: 3

8


Option: 4

9


Answers (1)

best_answer

Two sides of a triangle 5 cm and 12 cm.

Let a = 5 cm and b = 12 cm

Let the third side be x cm

\begin{array}{l} \therefore 12-5<x<12+5 \\ \Rightarrow 7<x<17 \end{array}

therefore, possible integer values for x are 8, 9, 10, 11, 12, 13, 14, 15 and 16.

\begin{aligned} &\text { Case } 1 \text { : If } b \text { is the longest side then } b^{2}>a^{2}+x^{2}\\ &\Rightarrow 12^{2}>5^{2}+x^{2}\\ &\Rightarrow 144-25>x^{2}\\ &\Rightarrow x^{2}<119\\ &\therefore x \text { can be } 8,9 \text { or } 10 . \end{aligned}

\begin{aligned} &\text { Case } 2: \text { If } x \text { is the longest side, then } x^{2}>a^{2}+b^{2}\\ &\Rightarrow x^{2}>5^{2}+12^{2} \Rightarrow x^{2} \Rightarrow 169\\ &\therefore x \text { can be } 14,15 \text { or } 16 \end{aligned}

The number of possible triangles = 6 (?The measurement third side is an integer in cm).

Posted by

shivangi.bhatnagar

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