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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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in Mathematics
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A. 6

B. 9

C. 10

D. 8

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Best answer

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

another triangle

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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