Given: △abc, m∠a=60°,
m∠c=45°, ab=9
find: perimeter of △abc,
area of △abc

Step-by-step explanation:

Drawing an altitude (BD) from B to AC can be very helpful. It divides this triangle into two that have side ratios you recognize.

A 30°-60°-90° triangle has side ratios 1 : √3 : 2, and a 45°-45°-90° triangle has side ratios 1 : 1 : √2.

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The given side of 9 is the longest side of a 30-60-90 triangle, and the side AD is the shortest, so is 1/2 of 9, or 4.5. Side BD is the intermediate length of that triangle, so is 4.5√3.

The other leg of triangle BDC is the same length, so DC = 4.5√3. The hypotenuse of that triangle, BC, is √2 times that, or 4.5√6.

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Now, we know the three sides are 9, (4.5+4.5√3), and 4.5√6. The perimeter is the sum of these, or ...

  P = 9 + 4.5 + 4.5√3 + 4.5√6

  P = 4.5(3 +√3 +√6) ≈ 32.3169

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The area is given by the formula ...

  A = 1/2bh

where the base length (b) is 4.5(1+√3), and the height is 4.5√3. Putting these values into the formula gives ...

  A = 1/2(4.5(1+√3))(4.5√3)

  A = 10.125(3+√3) ≈ 47.9120

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