Question

(ii) In the figure, ZABC = 90°, AB = 3x, BC = 4x and AC-30, then

(a) Using Pythagoras theorem, determine the value of x.
(b) Find the length of segments AB and BC.
(c) Find A (AABC).​

Answer 1

Answer:

(a) The value of x is 6

(b) The length of AB is 18 units

     and the length of BC is 24 units

(c) The area of ΔABC  is 216 units²

Step-by-step explanation:

In the given ΔABC,

∠ ABC = 90°

AB = 3X

BC = 4X

AC = 30

(a) According to Pythagoras Theorem,

Hypotenuse² = Base² + Perpendicular²

Substituting the values, we get,

    30² = (4x)² + (3x)²

=> 900 = 16x² + 9x²

=> 900 = 25x²

=> x² = 900/25

=> x² = 36

=> x = ± 6

Since here, x is a multiplying factor for the length and length cannot be negative.

Therefore, x = 6

(b) We need to find the length of the AB and BC

As we know that AB = 3x = 3*6 = 18

As we know that BC = 4x = 4*6 = 24

Therefore, the length of AB is 18 units

and the length of BC is 24 units

(c) We need to find the area of the ΔABC

We know that for aright angled triangle,

Area = ($\frac{1}{2}$) * Base * Height

Substituting the values,

Area = ($\frac{1}{2}$) * Base * Height

        = ($\frac{1}{2}$) * BC * AB

        = ($\frac{1}{2}$) * 24 * 18

        = 12 * 18

        = 216 units²

Therefore, the area of ΔABC  is 216 units²

Answer 2

Answer:

(a).  THE VALUE OF "x" IS 6.

(b). THE LENGTH OF SEGMENT AB IS 18 units AND BC IS 24 units.

(c).  THE AREA OF ΔABC IS 216 units².

Step by Step explanation:

• In context to the given question we have to find the value of "x", length of segment and area of triangle ABC

• given,

∠ ABC = 90°

AB = 3x

BC = 4x

Length of AC = 30

(a). Using Pythagoras Theorem, we know that

             a² = b² + c²

             where : a = hypotenuse , b= base and c = height

            Substituting the values, we get,

             30² = (4x)² + (3x)²

             900 = 16x² + 9x²

            900 = 25x²

            BY TRANSPOSING METHOD

             x² = 900/25

             x² = 36

             x = ± 6

             (as we know length can never be negative )

            ∴ x = 6

SO,THE VALUE OF "x" IS 6

• (b) We need to find the length of the AB and BC

            given:

            AB = 3x

            BC = 4x

            Now by putting the value of x we found

            AB = (3) (6) = 18 units

            BC = (4) (6) = 24 units

SO, THE LENGTH OF SEGMENT AB IS 18 units AND BC IS 24 units.

• (c) We need to find the area of the ΔABC

             We know that

            Area of a triangle = (1/2) x base x height

            By putting the value of known value we get,

            Area of a triangle = (1/2) x BC x AB

            Area of a triangle = (1/2) x 24x 18      

            Area of a triangle = (24x18)/2

            Area of a triangle = 432/2

            Area of a triangle = 216 units²

   

∴ THE AREA OF ΔABC IS 216 units²