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I'm not completely sure what your question is asking, but the following information should be helpful.
You know triangle ABC is a right triangle, and that angle A is the right angle (since it is opposite the hypotenuse BC). You also know that angle C is 60 degrees. So ABC is a 30-60-90 triangle (the sum of the angles has to be 180, and 180 - (90 + 60) = 30).
Since AD is the altitude from A to BC, you know by definition that the angles on either side of the altitude are right angles.
Now, let's look at the smaller triangles.
ADC is a right angle triangle (angle D of that triangle is one of the right angles made by the altitude AD). You also know angle C is 60 degrees. Again, then, this is a 30-60-90 triangle.
You can do the same analysis for small triangle DBA. D is a right angle, B is a 30 degree angle (see paragraph 2 above), and so again we have a 30-60-90 triangle.
One of the rules of similarity is that if 2 angles of a triangle are equal to 2 angles of another triangle, the triangles are similar.
ABC is similar to ADC because the right angles and angle C of both are equal. ABC is similar to DAB because the right angles and angle B of both are equal.
ABC is a right angled triangle with a right angle at A . AD is the altitude from A to BC from A.
Therefore, the area of the triangle ABC = (1/2)AC*AB (1)
In triangles ABC and ADC,
AngleA = angleADC and Angle C is common to both. The other angle has to be equal. Therefore the triangles are |||r. Therefore, the corresponding sides should bear the same ratio:
DC= AC^2/BC; AD= AC*AB/BC.
DC*AD = (AC^3)(AB)/BC^2
area of triangle ADC=
From (1) and (2),
Area of triangleADC/Area of triangle ABC=
In triangle ABC and ADB: angle A=angle ADC right angles, angle B is common. Therefore the triagles are |||r. So,
AD=(AB)(AC) /BC and BD= AB^2/BC.
Therefore, Area of
triangle ADC = (1/2)AD*DC= (AB^3)(AC)/BC^2 (3)
From (1) and (3),
triangle ABC/ triangle ABD=BC^2/AB^2
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