Overlapping figures

CF = (4)^0.5 = 2

For triangle ABF, the altitude AF = CF(3)^0.5 = 2*(3)^0.5
Since it is a right angle isosceles triangle, AB = base = CF
Area
= (base * height) / 2
= (CF)*(CF)/2
= [(2*(3)^0.5)^2] / 2
= 6 square units

[spoiler](E)[/spoiler] is the correct option

Hi there thanks so much for your reply,
are the two triangles similar?

because i managed to get that the left triangles sides would be
2, root 3 (hypoteneuse), and 1 (is this right)
the squares side would be 2

by altitude u mean height right?

why are we multiplying the 2(CF) *root 3 to get the length (what formula is this)

i understand that ABF is an isoceles triangle...and the ratio of sides should be 1:1:root 2 right?

would you mind going into a little more detail
?
thank you so much
marianne

Hi Marianne,

No, the triangles are not similar.
And yes, 'altitude' and 'height' are the same thing.

In a right angled triangle in which the angles are 90 degrees, 60 degrees and 30 degrees, the ratio of sides is a : b : c = 2 : (3)^0.5 : 1



a : b : c = 2 : (3)^0.5 : 1 is a concise way of writing this:
(i) a : b = 2 : (3)^0.5
(ii) b : c = (3)^0.5 : 1
(iii) c : a = 1 : 2

Note that:

(i) the side opposite to the highest angle (90 deg) is longest and the side opposite to the smallest angle (30 deg) is the shortest.

(ii) This set of ratios is only for a 90-60-30 triangle. Remember them, so that we don't need to go into trignometry.

(iii) These are just ratios. They don't mean that a is 2 units, b is (3)^0.5 units and c is 1 units.

Can you now answer/explain:
(a) how we will apply these ratios in the triangle ABF of your question?
(b) What the area of right angled triangle ABC, with angle B = 90 deg and angle C = 60 deg, and hypotenuse = 6 units?

Thank you very much