Question

if triangle PQR is right angled at Q, PQ = QR if QS perpendicular to PR and QS = 4cm what is the perimeter of triangle PQR​ is ? dont copy paste the answer

Answer 1

Answer:

The given Question is solved in the uploaded photos

Answer 2

$\large\underline{\sf{Given- }}$

A triangle PQR, right angled at Q

PQ = QR

QS perpendicular to PR

QS = 4cm

$\large\underline{\sf{To\:Find - }}$

What is the perimeter of triangle PQR ?

$\large\underline{\sf{Solution-}}$

GIVEN THAT,

A triangle PQR, right angled at Q such that PQ = QR.

So, Let assume that PQ = QR = x cm

Now, In triangle PQR

By using Pythagoras Theorem, we have

$\rm :\longmapsto\: {PR}^{2} = {QR}^{2} + {PQ}^{2}$

$\rm :\longmapsto\: {PR}^{2} = {x}^{2} + {x}^{2}$

$\rm :\longmapsto\: {PR}^{2} = 2{x}^{2}$

$\bf\implies \:PR = \sqrt{2}x \: cm$

Now,

$\rm :\longmapsto\:Area_{\triangle PQR} = \dfrac{1}{2} \times PQ \times QR$

$\bf\implies \:Area_{\triangle PQR} = \dfrac{1}{2} \times x \times x = \dfrac{ {x}^{2} }{2} - - (1)$

Also,

$\rm :\longmapsto\:Area_{\triangle PQR} = \dfrac{1}{2} \times QS \times PR$

$\rm :\longmapsto\:Area_{\triangle PQR} = \dfrac{1}{2} \times 4 \times \sqrt{2} x$

$\bf\implies \:Area_{\triangle PQR} = 2 \sqrt{2}x \: cm - - - (2)$

So, From equation (1) and (2), we have

$\rm :\longmapsto\:2 \sqrt{2}x = \dfrac{ {x}^{2} }{2}$

$\bf\implies \:x = 4 \sqrt{2} \: cm$

Hence, we have

$\rm :\longmapsto\:PQ = x = 4 \sqrt{2} \: cm$

$\rm :\longmapsto\:QR = x = 4 \sqrt{2} \: cm$

$\rm :\longmapsto\:PR = \sqrt{2} x = \sqrt{2} \times 4 \sqrt{2} = 8 \: cm$

So,

$\rm :\longmapsto\:Perimeter_{\triangle PQR} = PQ + QR + PR$

$\rm :\longmapsto\:Perimeter_{\triangle PQR} = 4 \sqrt{2} + 4 \sqrt{2} + 8$

$\rm :\longmapsto\:Perimeter_{\triangle PQR} = 8\sqrt{2} + 8$

$\rm :\longmapsto\:Perimeter_{\triangle PQR} = 8(\sqrt{2} + 1) \: cm$

Additional Information :-

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem,

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.