Solve the following (Any One) : If a line is drawn parallel to one side of a triangle and intersects the other two sides, then the other two sides are divided in the same ratio. OR In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Answers
Answer:
if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Step-by-step explanation:
ANSWER
Given:- ABC is a triangle
AC
2
=AB
2
+BC
2
To prove:- ∠B=90°
Construction:- Construct a triangle PQR right angled at Q such that, PQ=AB and QR=BC
Proof:-
In △PQR
PR
2
=PQ
2
+QR
2
(By pythagoras theorem)
⇒PR
2
=AB
2
+BC
2
.....(1)(∵AB=PQ and QR=BC)
AC
2
=AB
2
+BC
2
.....(2)(Given)
From equation (1)&(2), we have
AC
2
=PR
2
⇒AC=PR.....(3)
Now, in △ABC and △PQR
AB=PQ
BC=QR
AC=PR(From (3))
∴△ABC≅△PQR(By SSS congruency)
Therefore, by C.P.C.T.,
∠B=∠Q
∵∠Q=90°
∴∠B=90°
Hence proved.