Question

The figure shows a right-angled triangle ABC with dimensions as shown. (1) If the perimeter of the triangle is 17 cm, write down an expression, in terms of x, for the length of AB. (1Hence, formulate an equation in x and show that it simplifies to 2x - 18x + 17 = 0. x
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Answer 1

Answer:

The value of $x$ is 1.072.

Step-by-step explanation:

Given a right angled triangle $ABC$. Since perimeter is 17, we have

$AB+BC+AC=17$

Here $BC=x\ cm, AC=8cm$. So

$AB=17-BC-AC=17-x-8=9-x$.

Now using Pythagoras theorem,

$AC^2=AB^2+BC^2\\8^2=(9-x)^2+x^2\\64=81-18x+x^2+x^2\\2x^2-18x+81-64=0\\2x^2-18x+17=0$

Value of x is obtained by solving the above equation.

$x=\dfrac{18\pm\sqrt{(-18)^2-4\times 2\times 17} }{2\times 2}\\=\dfrac{18\pm\sqrt{324-136} }{4}=\dfrac{18\pm\sqrt{188} }{4}\\=\dfrac{9\pm\sqrt{47} }{2}=7.928,1.072$

Hence area of the triangle is

$\frac{1}{2}\times 7.928\times 1.072=4.249cm^2$

Answer 2

Tip:

• Perimeter of triangle = Sum of all three sides.
• Area of triangle = $\frac{1}{2}\times base \times height$

Explanation:

• In the given figure we have $AC=8~cm\\BC=x~cm$ and perimeter of triangle is $17~cm$.
• We have to find the side $AB$ in terms of $x$, and simplify the given equation and area of triangle.
• We will solve the question by using the formula of perimeter and by Pythagoras theorem and Area of triangle.

Steps:

Step 1 of 4:

(i) Perimeter of triangle = $AB+AC+BC$

$17=AB+8+x$

$AB=17-8-x\\AB=9-x$

Step 2 of 4:

(ii) By Pythagoras Theorem:

$AB^2+BC^2=AC^2$

Putting the value of $AB,BC,AC$ in the above equation,

$(9-x)^2+x^2=8^2\\81+x^2-18x+x^2=64\\2x^2-18x+81-64=0\\2x^2-18x+17=0$

This is the required equation.

Step 3 of 4:

Now, solving this equation,

$2x^2-18x+17=0\\x=\frac{18\pm \sqrt{(18)^2-(4)(2)(17)}}{2\times2}\\x=\frac{18\pm \sqrt{324-136}}{4}\\x=\frac{18\pm \sqrt{188}}{4}$

$x=\frac{9\pm\sqrt{47}}{2}\\x=\frac{9\pm6.855}{2}$

So, $x=\frac{9+6.85}{2},\frac{9-6.85}{2}\\x=7.925,1.075$

Step 4 of 4:

Area of Triangle = $\frac{1}{2}\times base \times height$

                           $=\frac{1}{2}\times AB\times BC\\=\frac{1}{2}(7.925)(1.075)\\=\frac{8.519}{2}\\=4.2595 ~ cm^2$

Final Answer:

So, Area of triangle $=4.2595~cm^2$