arrow_back A triangle has interior angles with measures in the ratio of 1:2:3. If the longest side length of the triangle is 18, what is the length of the shortest side?

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A. 6

B. 9

C. 10

D. 8

Since the sum of all interior angles of a triangle add up to $180^{\circ}$ then we can determine the size of each angle as follows:

Let $x; 2x;3x$ be the three interior angles. So,

$x+2x+3x=180$

Solving the equation:

$x=30^{\circ}$

So our 3 angles are $30^{\circ}; 60^{\circ}; 90^{\circ}$

Since one of the angles is $90^{\circ}$ it means this is a right-angled triangle as shown below

From the triangle we can formulate these equations using the SIne Rule.

$\dfrac{b}{\sin{60^{\circ}}}=\dfrac{a}{\sin{30^{\circ}}}=\dfrac{18}{\sin{90^{\circ}}}$

So

$\dfrac{b}{\sin{60^{\circ}}}=\dfrac{18}{\sin{90^{\circ}}}$

bycross-multiplication, $b=\dfrac{18\sin{60^{\circ}}}{\sin{90}} = 15.588$

and

$\dfrac{a}{\sin{30^{\circ}}}=\dfrac{18}{\sin{90^{\circ}}}$

bycross-multiplication, $a=\dfrac{18\sin{30^{\circ}}}{\sin{90}} = 9$

Therefore the shortest side is 9 units long.

by Platinum
(104,456 points)

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