In a triangle ABC, AD is perpendicular to BC. D divides BC in the ratio 1:3 internally. Find BC, if AB = 9cm and AC = 21cm. (A) 12√5cm (B) 15√5cm (C) 16√5cm (D) 18√5cm Correct answer is A. Can you explain this question?

Question

Answer

Problem: In a triangle ABC, AD is perpendicular to BC. D divides BC in the ratio 1:3 internally. Find BC, if AB = 9cm and AC = 21cm.

Solution:
To solve this problem, we will use the Pythagorean Theorem and the concept of similar triangles. Let's start by drawing a diagram of the triangle ABC:

Step 1: Find the length of AD:
Since AD is perpendicular to BC, we can use the Pythagorean Theorem to find its length. Let's call the length of AD as x. Then, we have:

x^2 + BD^2 = AB^2
x^2 + (BC - CD)^2 = AB^2
x^2 + (BC - 3x)^2 = 9^2

Step 2: Find the length of CD:
Since D divides BC in the ratio 1:3 internally, we can write:

CD = 3x

Step 3: Use similar triangles to find BC:
Let's draw a line parallel to AC passing through point D. This line divides triangle ABC into two similar triangles ABD and ACD:

Using the concept of similar triangles, we can write:

AB/BD = AC/CD
9/BD = 21/3x
BD = 3x/7

Now, we can use the Pythagorean Theorem in triangle ABD to find BD:

BD^2 + x^2 = 9^2
(3x/7)^2 + x^2 = 81
10x^2/49 = 81
x^2 = 81*49/10
x = 9√5/2

Step 4: Find BC:
We can use the equation we derived in Step 1 to find BC:

x^2 + (BC - 3x)^2 = 9^2
(9√5/2)^2 + (BC - 27√5/2)^2 = 81
81*5/4 + BC^2 - 27*9√5/2*BC + 27^2*5/4 = 81
BC^2 - 27*9√5/2*BC + 27^2*5/4 = 0
Solving this quadratic equation, we get:

BC = 12√5cm

Therefore, the answer is (A) 12√5cm.

Answer

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