Question

Find the measures of the marked angles in the triangle below if you know that a/c=10/11 and a/2=b/3. Hint: if x/y=w/z then xz=wy. In other words, if two fractions are equal, the products you obtain from cross-multiplying will also be equal. (Do not use a protractor to measure the angles below since the diagram is not drawn to perfectly match the angles of your answer.)

Answer 1

Solution:-

Let the side a is equal to x .

So, a = x .....(1)

Since a/c = 10/11

So, c= (11/10)a = 1.1a = 1.1x ....(2)

Since a/2 = b/3

So, b = 3a/2 = 1.5a = 1.5x ....(3)

By using above equations, we get

s = (a+b+c)/2 = (x +1.5x +1.1x)/2 = 3.6x/2 = 1.8x

Now, the three sides a, b and c corrosponding to angles A,B and C respectively arecshown in the figure below-

Now, we know that

On putting the values, we get

This implies

Or A = 20.9×2 = 41.8°

So, angle A = 41.8° .....(1)

Similarly,

On putting the values, we get

This implies

Or B = 45.52×2 = 91.04°

So, angle B = 91.04° .....(5)

Now, angle C can be find out as

C = 180° - (A + B)

C= 180° - (41.8° + 91.04°)

C = 180° -132.84°

C = 47.16° ....(6)

From equations (4), (5), (6), we get angles as

A = 41.8°

B = 91.04°

C = 47.16°

Hence, angles are A = 41.8°, B = 91.04°, C = 47.16° .