This question was uploaded on 12/08/21 on social media accounts.
A Right Angled Triangle is given. A square is drawn in that triangle with one side on the hypotenuse. Find the length of the side of the square when sides of a triangle are given 3-4-5.
I given 3 solutions simultaneously. You may go for any one of these solutions.
Step 1:
Give Names to all vertices.
Let side of square be x.
Sides are given 3, 4, 5.
Step 2:
Observe triangles ABC, ASP, PBQ, QRC:
All have one angle 90°.
And other two angle are also similar.
⇒ △ABC ~ △ASP ~ △PBQ ~ △QRC
Consider △ABC ~ △ASP:
\[\color{orange} {PS = x}\]
\[\color{blue} {AS = \frac{4}{3}x}\]
\[\color{green} {AS = \frac{5}{3}x}\]
Consider △ABC ~ △PBQ:
\[\color{orange} {PQ = x}\]
\[\color{green} {PB = \frac{4}{5}x}\]
\[\color{purple} {BQ = \frac{3}{5}x}\]
Consider △ABC ~ △QRC:
\[\color{orange} {QR = x}\]
\[\color{purple} {QC = \frac{5}{4}x}\]
\[\color{blue} {BQ = \frac{3}{4}x}\]
Step 3: (You may go with any one of following calculations)
\[\color{green} {AB = \frac{5}{3}x+\frac{4}{5}x}\]
\[\color{green} {\Rightarrow 4= \frac{37}{15}x}\]
\[\color{green} {\Rightarrow x= \frac{60}{37}}\]
\[\color{purple} {BC = \frac{3}{5}x+\frac{5}{4}x}\]
\[\color{purple} {\Rightarrow 3= \frac{37}{20}x}\]
\[\color{purple} {\Rightarrow x= \frac{60}{37}}\]
\[\color{blue} {AC = \frac{4}{3}x+x+\frac{3}{4}x}\]
\[\color{blue} {\Rightarrow 5= \frac{37}{12}x}\]
\[\color{blue} {\Rightarrow x= \frac{60}{37}}\]
⇒ x = ⁶⁰⁄₃₇
Image Solution:
Puzzle related to Geometry, Triangle, Square
