Question

the points A(4,7), B(p,3) and C(7,3) are the vertex of a right Triangle a right triangle at B find the value of p ​

Answer 1

Given: The points A (4, 7), B (p, 3) and C(7, 3) are the vertex of a right angle triangle.

To find: The value of p.

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀

▪︎ As it is a right angle triangle, we can use Pythagoras theorem.

✇ Now, By using Pythagoras theorem in right angle triangle. Let's find out the value of p —

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$\star\:{\underline{\boxed{\pmb{\sf{\Big(AC\Big)^2 = \Big(AB\Big)^2 + \Big(BC\Big)^2}}}}}$

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$\dashrightarrow\sf \Big(7 - 4\Big)^2 + \Big(3 - 7\Big)^2 = \Big(p - 4\Big)^2 + \Big(3 - 7 \Big)^2 + \Big(p - 7\Big)^2 + \Big(3 - 3\Big)^2 \\\\\\\dashrightarrow\sf 3^2 + 4^2 = \Big(p - 4\Big)^2 + 4^2 + \Big(p - 7\Big)^2\\\\\\\dashrightarrow\sf 9 + \cancel{16} = \Big(p - 4\Big)^2 + \cancel{16} + \Big(p - 7\Big)^2\\\\\\\dashrightarrow\sf 9 = p^2 + 16 - 8p + p^2 + 49 - 14p\\\\\\\dashrightarrow\sf 2p^2 - 22p + 56 = 0\\\\\\\dashrightarrow\sf p^2 - 11p + 28 = 0\\\\\\\dashrightarrow\sf p^2 - 7p - 4p + 28 = 0\\\\\\\dashrightarrow\sf p(p - 7) - 4(p - 7) = 0\\\\\\\dashrightarrow\sf (p - 4) (p - 7) = 0\\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{\pink{p = 4 \;or\; 7}}}}}\;\bigstar$

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⠀⠀⠀⠀⠀$\therefore{\underline{\textsf{Hence, the required value of p are \textbf{4} \sf{and} \textbf{7}.}}}$

Answer 2

$\huge\mathbb\fcolorbox{purple}{lavenderblush}{✰Question✰}$

The points A(4,7), B(p,3) and C(7,3) are the vertex of a right Triangle a right triangle at B find the value of p ​

$\huge \sf \fbox\orange{A}\fbox\pink{n}\fbox\blue{s}\fbox\red{w}\fbox \purple{e}\fbox\red{r}$

The value of  can be 4 or 7

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{(Step By StepExplanation}}}}}}\red\bigstar\:\: \\ \end{gathered}$

$\Delta$ ABC is right angled at B.

$\blue{AC {}^{2} = AB {}^{2} + BC {}^{2} \: .(1)}$

$\blue{Also, \: A = (4,7) \:,\: B = (p,3) \: and \: C = (7,3)}Also,A=(4,7),B=(p,3)andC=(7,3)$

$\green{Now, \: AC {}^{2} = (7 - 4) {}^{2} + (3 - 7) {}^{2} = (3) {}^{2} + ( - 4) {}^{2} = 9 + 16 = 25}$

$\red{AB {}^{2} = (p - 4) {}^{2} + (3 - 7) {}^{2} = p {}^{2} - 8p + 16 + ( - 4) {}^{2}}$

$=\orange{p {}^{2} - 8p + 16 + 16}$

$=\orange{ p {}^{2} - 8p + 32}$'

$=\green{p {}^{2} - 14p + 49}$

$\green{From \: (1), \: We \: have}$

$\blue{25 = (p {}^{2} - 8p + 32) + (p {}^{2} - 14p + 49)}$

$\red{25 = 2p {}^{2} - 22p + 81}$

$= \orange{ 2p {}^{2} - 22p + 56 = 0}$

$= \blue{p {}^{2} - 11p + 28 = 0}$

$= \blue{p {}^{2} - 7p - 4p + 28 = 0$

$=\orange{ p(p - 7) - 4(p - 7) = 0}$

$=\green{ (p - 7) \: (p - 4) = 0}=(p−7)(p−4)=0$

$=\red{p = 7 \: and \: p = 4}$

Hence the answer is $\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{The }}}}}}\red\bigstar\:\: \\ \end{gathered}$$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{Value }}}}}}\red\bigstar\:\: \\ \end{gathered}$$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{of can }}}}}}\red\bigstar\:\: \\ \end{gathered}$$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{Be }}}}}}\red\bigstar\:\: \\ \end{gathered}$ $\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{4 or 7 }}}}}}\red\bigstar\:\: \\ \end{gathered}$