Question

construct an equaliteral triangle such that a given point inside it is distant 2 units from one vertex

Answer 1

The answer is 6.85.

Consider, x-y axis, and side of equilateral triangle is ' a '.

Given,

$\begin{aligned}&A P=3 \text { units } \\&B P=2 \text { units } \\&C P=4 \text { units }\end{aligned}$

Let x and y are point in x-y axis.

Using, distance formula here

$A P^{2}=(x-0)^{2}+(y-0)^{2}$

$3^{2}=x^{2}+y^{2}$

$x^{2}+y^{2}=$__________(i)

$p c^{2}=(x-a)^{2}+(y-0)^{2} 4^{2}=x^{2}-2 a x+a^{2}+y^{2}$

from (i)

$16=x^{2} 9-2 a x+a^{2}$

$2 a x=a^{2}+9-16$

$\left[x=\frac{\left(a^{2}-7\right)}{2 a}\right]$

$B p^{2}=\left(x-\frac{a}{2}\right)^{2}+\left(y-\frac{\sqrt{3}}{2} a\right)^{2}$

$2^{2}=x^{2}-a x+\frac{a^{2}}{4}+y^{2}-\sqrt{3} a b y+\frac{3}{4} a^{2}$

$4=\left(x^{2}+y^{2}\right)+\frac{a^{2}+3 a^{2}}{4}-a\left(\frac{a^{2}-7}{2 a}\right)-\sqrt{3} a y$

from (i)

$4=9+a^{2}-a^{2}+7-\sqrt{3} a y$

$\sqrt{3} a y=16-4$

put x and y in equarion_______(i)

$\left(\frac{12}{\sqrt{3} a}\right)^{2}+\left(\frac{a^{2}-7}{2 a}\right)^{2}=9$

$\frac{144}{3 a^{2}}+\frac{\left(a^{2}-7\right)^{2}}{4 a^{2}}=9$

$144 \times 4+3\left(a^{2}-7\right)^{2}=12 \times 9 a^{2}$

$576+3\left(a^{2}-7\right)^{2}=108 a^{2}$

$put a^{2}=t 576+3(t-7)^{2}=108 t$

$\begin{aligned} 576+3\left(t^{2}-14 t+49\right) &=108 t \\ 576+3 t^{2}-42 t+147 &=108 t \end{aligned}$

$3 t^{2}-150 t+429=0 t^{2}-50 t+143=0 t=50 \pm \sqrt{50^{2}-4 \times 143}$

$t=\frac{50 \pm \sqrt{1928}}{2}t=\frac{50 \pm 43.91}{2}$

$t=\frac{50 \pm 43.91}{2}\\t=46.9545, \quad t=3.0455$

$\begin{array}{ll}a^{2}=46.9545, \\ a=\sqrt{46.9545}, & a^{2}=3.0455 \\ a, a=1.745\end{array}$

$a=6.85$

for this point P is outside Therefore, it is not consider as solution (because in question inside point ls mention)

$a=6.85 units$

What is equilateral triangle?

• A triangle with the same length on all three sides
• If all of the sides of a form are the same length, it is said to be equilateral. Many forms, such as triangles and squares, are covered in geometry instruction.
• Because all of the sides of a square have the same length, it is equilateral.

To learn more about equilateral triangle visit:

brainly.com/question/3461022

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