In triangle ABC, CD perpendicular AB CA = 2 ad and BD = 3AD. prove that Angle BCA = 90 degree
To prove that angle BCA is 90 degrees, we need to use the given information about the triangle and apply some geometric concepts.
Let's break down the steps to prove this:
Step 1: Draw the triangle and label the given points and lengths.
C
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D /__| A
B
We have triangle ABC with points A, B, and C. CD is perpendicular to AB, and CA is labeled as 2ad. BD is labeled as 3AD.
Step 2: Identify any relevant concepts or theorems that could be applied.
To prove that angle BCA is 90 degrees, we can utilize the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Step 3: Apply the relevant concepts or theorems to the given information.
In our triangle, we can consider triangle BCD to be a right triangle with CD as the hypotenuse. Triangle BCD and triangle ABC share the same height, which is AD.
According to the Pythagorean theorem, we know that,
(BD)^2 + (CD)^2 = (BC)^2.
Substituting the given values,
(3AD)^2 + (2AD)^2 = (BC)^2.
Simplifying this equation,
9(AD)^2 + 4(AD)^2 = (BC)^2.
Combining the terms,
13(AD)^2 = (BC)^2.
Step 4: Draw a conclusion based on the obtained equation.
From the equation 13(AD)^2 = (BC)^2, we can conclude that the square of the length of BC is equal to 13 times the square of the length of AD.
Step 5: Use the conclusion to prove the desired result.
Since the square of any real number is always positive, if 13(AD)^2 = (BC)^2, both sides of the equation must be positive.
This means that both BC and AD have a positive length. Therefore, the length of BC cannot be negative, and we can conclude that BC is greater than zero.
If BC is greater than zero, then (BC)^2 is also greater than zero.
In the equation 13(AD)^2 = (BC)^2, since both sides are positive and (BC)^2 > 0, we can observe that no matter what value the length of AD takes, (BC)^2 will always be greater than 0.
However, for BC to be greater than 0 and (BC)^2 to be greater than 0, it is necessary for (AD)^2 to also be greater than 0.
The only way for a real number to have a positive square is if the real number itself is positive. Therefore, (AD)^2 > 0, which means that AD must also be greater than 0.
With AD and BC both greater than 0, we have fulfilled the condition for the Pythagorean theorem to hold true.
According to the Pythagorean theorem, the lengths of the sides in a right triangle satisfy the equation (BD)^2 + (CD)^2 = (BC)^2.
However, we know that (AD)^2 > 0 and (BC)^2 > 0, which means that the only way for the equation to hold true is if (BD)^2 + (CD)^2 = (BC)^2.
Since (BD)^2 + (CD)^2 = (BC)^2, we have proved that triangle BCD is a right triangle, and angle BCA is equal to 90 degrees.
Therefore, angle BCA = 90 degrees.