Question

Q34.

ABC is a triangle.
Angle ABC= angle BCA.
The length of side AB is (3x - 5) cm.
The length of side AC is (19 - x) cm.
The length of side BC is 2x cm.
Work out the perimeter of the triangle.
Give your answer as a number of centimetres.

Answer 1

To calculate the perimeter of triangle ABC, the value of x is determined by solving the equation 3x - 5 = 19 - x, yielding x = 6. Substituting x into the given expressions for the sides of the triangle, the lengths become AB = AC = 13 cm and BC = 12 cm, resulting in a total perimeter of 38 cm.

Step-by-step explanation:

To find the perimeter of triangle ABC, we need to add together the lengths of all three sides. Since angle ABC is equal to angle BCA, the triangle is isosceles and sides AB and AC are of equal length, so we can set (3x - 5) equal to (19 - x) to solve for x. Solving the equation:

3x - 5 = 19 - x
4x = 24
x = 6

Now we can find the lengths of each side:

• AB = 3x - 5 = 3(6) - 5 = 18 - 5 = 13 cm
• AC = 19 - x = 19 - 6 = 13 cm
• BC = 2x = 2(6) = 12 cm

Finally, to find the perimeter, we add the lengths:

Perimeter = AB + AC + BC = 13 cm + 13 cm + 12 cm = 38 cm

Answer 2

⇒ Perimeter is the sum of all the sides.

⇒In Triangle ABC it means we are going to add all the lengths of the sides

⇒ P=AB+BC+AC

Given that AB is (3x-5)cm, AC is (19-x) and BC is (2x)cm

⇒ P=[(3x-5)+(2x)+(19-x)]cm

⇒Simplify by grouping like terms together.

`P=[3x+2x-x-5+19]cm\\P=(4x+14)cm`

If it is given that angle ABC = angle BCA , this implies that side AB= AC, by reason "Sides opposite equal angles". ∴ 3x-5=19-x,this will help us find the values of x and the numerical value of the perimeter.

⇒Solving for x we have:

`3x-5= 19-x\\3x+x=19+5\\4x=24\\(4x)/(4) =(24)/(4) \\x=6`

⇒From the information in the question the value of x is 6,now we can find the numerical values of the Perimeter by substituting the value of x which is 6 in P=(4x+14)cm

∴
`P=[4(4)+14)cm\\P=(16+14)cm\\P=30cm`

∴The Perimeter of the Triangle ABC is 30cm