Question

If the area of triangle is 7 sq.unit then find the value ok k by using determinant method if the vertices of triangle are
A(4,9),B(K, 0) and C(4,K) ​

Answer 1

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✤ Required Answer:

K = 2,11

✒ Given that :

Area of triangle is 7 sq.unit

vertices of triangle are

A(4,9),B(K, 0) and C(4,K)

✒ we hv to find:

value of k by using determinant method

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✤ How to Solve?

Firstly u HV to know that determinant is the scalar value which is computed from different elements of a square matrix that has certain properties of a linear transformation. Let us now learn how to use the determinant to find the area of a triangle. Let’s say that (x1, y1), (x2, y2 ), and ( x3, y3 ) are three points of the triangle in the cartesian plane. Now the area of the triangle of the will be given as: 

$k = ½ [ x1 ( y2 - y3 ) + x2 ( y3 - y1 ) + x3 ( y1 - y2 ) ]$

Here, k is the area of the triangle using determinant and the vertices of the triangle are represented by (x1, y1), (x2, y2 ), and ( x3, y3 ).

U should also know how to solve quadratic equation..

ツ Now let's solve the question by our knowledge.

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✤ Solution:

According to formula,

$k = ½ [ x1 ( y2 - y3 ) + x2 ( y3 - y1 ) + x3 ( y1 - y2 ) ]$

7 = ½ [4 ( 0-K ) + K ( K - 9 ) + 4 (9 - 0)]

7 = ½ [ -4K + K^2 - 9K +36 ]

14 = K^2 - 13K +36

K^2 -13K +22 = 0

So, Let's solve the quadratic equation,

Factoring k2-13k+22

The first term is, k2 its coefficient is 1 .

The middle term is, -13k its coefficient is -13 .

The last term, "the constant", is +22

Step-1 : Multiply the coefficient of the first term by the constant 1 • 22 = 22

Step-2 : Find two factors of 22 whose sum equals the coefficient of the middle term, which is -13 .

-22 + -1 = -23

-11 + -2 = -13 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -11 and -2

k2 - 11k - 2k - 22

Step-4 : Add up the first 2 terms, pulling out like factors :

k • (k-11)

Add up the last 2 terms, pulling out common factors :

2 • (k-11)

Step-5 : Add up the four terms of step 4 :

(k-2) • (k-11)

Which is the desired factorization

So K = 2,11

☀️ Hence, solved !!

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