Question

In the right △ABC with m∠C=90°, m∠B=75°, and AB=12 cm. Find the area of △ABC. without trigonometry! worth 49 points

Answer 1

Answer: 18 square cm

Explanation:

*picture very note to scale im sorry lol*

Given: △ABC, m∠C=90°, m∠B=75°, AB=12 cm

m∠A = 15° (sum of all angles in a triangle is 180 degrees and 180-90-75=15)

CM - median to hypotenuse

CL - altitude

m<CLB = m<CLM = 90 degrees (def of altitude)

CM=MA=MB=1/2 AB = 6 cm (median to hypotenuse theorem)

m<MBC=m<BCM=75 degrees (base angles theorem of iso triangle)

m<BMC = 30 degrees (sum of all angles in a triangle, 180-75-75 = 30)

m<LCM = 60 degrees (sum of all angles in a triangle of triangle LCM, 180-90-30=60)

so now we know that triangle LCM is a 30-60-90 triangle with a hypotenuse of CM (6 cm)

LC = 1/2 of MC = 3 cm (leg opposite to 30 degree <)

So now we know that the height (altitude) of the triangle is 3 cm and the length is 12 cm.

Then, we can find the area by doing the (height * length)/2 = (3*12)/2, which brings us to our answer of 18 cm.

lmk if you don't understand anything in here i'm happy to clarify!

hope this helps!

Answer 2

There might be two ways to go about this

(1) I am going to assume that we can construct a second (reference) triangle - and you confirmed that it is ok to use trigonometry on that, and then we use the relationship between areas of similar triangles to get what we want. I choose a triangle DEF with same angles, 15, 75, and 90 degrees, and the hypotenuse DE a of length 1 (that is a triangle similar to ABC). I use sin/cos to determine the side lengths: sin(15)=EF and cos(15)=DF and then compute the area(DEF) =EF*DF/2. This turns out to be 1/8 = 0.125.

Now one can use the area formula for similar triangles to figure out the area of ABC - this without trigonometry now: area(ABC)/area(DEF)=(12/1)^2

so area(ABC)=144*area(DEF)=144*0.125=18

(2) Construct the triangle ABC geometrically using compass, protractor, and a ruler. Draw a line segment AB of length 12. Using the compass draw a (Thales') semi-circle centered at the midpoint of AB with radius of 6. Then, using the protractor, draw a line at 75 degrees going from point B. The intersection with the semicircle will give you point C. Finally. draw a line from C to A, completing the triangle. Then, using ruler, measure the length BC and AC.

Calculate the area(ABC)=BC*CA/2, which should come out close to 18, if you drew precisely enough.