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Triangles Class 10 Notes For Maths Chapter 6

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Types of Triangles Timilar Tongruent Properties Different Angles

1. Similar Triangles:- Two triangles are said to be similar, if
(a) their corresponding angles are equal and
(b) their corresponding sides are in proportion (or are in the same ration).

2. Basic proportionality Theorem [ or Thales theorem ].

3. Converse of Basic proportionality Theorem.

4. Criteria for similarity of Triangles.
(a) AA or AAA similarity criterion.
(b) SAS similarity criterion.
(c) SSS similarity criterion.

5. Areas of similar triangles.

6. Pythagoras theorem.

7. Converse of Pythagoras theorem

1. ABC is a right-angled triangle, right-angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 6cm and 8 cm. Find the radius of the in circle.
(Ans: r=2)

TRIANGLES

right-angled triangle,

2. ABC is a triangle. PQ is the line segment intersecting AB in P and AC in Q such that PQ parallel to BC and divides triangle ABC into two parts equal in area. Find BP: AB.

Ans: Refer example problem of text book.

3. In a right triangle ABC, right angled at C, P and Q are points of the sides CA and CB respectively, which divide these sides in the ratio 2: 1. Prove that
9AQ2= 9AC2 +4BC2
9BP2= 9BC2 + 4AC2
9 (AQ2+BP2) = 13AB2


Similar Triangles

right triangle ABC,

4. P and Q are the mid points on the sides CA and CB respectively of triangle ABC right angled at C. Prove that 4(AQ2 +BP2) = 5AB2

Self Practice

5. In an equilateral triangle ABC, the side BC is trisected at D.
Prove that 9AD2 = 7AB2

Self Practice

6. There is a staircase as shown in figure connecting points A and B. Measurements of steps are marked in the figure. Find the straight distance between A and B.
(Ans:10)

connecting points

Ans: Apply Pythagoras theorem for each right triangle add to get length of AB.

7. Find the length of the second diagonal of a rhombus, whose side is 5cm and one of the diagonals is
6cm. (Ans: 8cm)

Ans: Length of the other diagonal
= 2(BO) where BO = 4cm

8. Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle.

Ans: To prove 3(AB2 + BC2 + CA2) = 4 (AD2+ BE2 + CF2)
In any triangle sum of squares of any two sides is equal to twice the square of half of third side, together with twice the square of medianbisecting it
.any triangle sum of squares

isosceles triangle

9. ABC is an isosceles triangle is which AB=AC=10cm.BC=12. PQRS is a rectangle inside the isosceles triangle. Given PQ=SR= y cm, PS=QR=2x. Prove

isosceles triangle


rectangle
rectangle inside

acute angled triangle

11. If ABC is an acute angled triangle , acute angled at B and prove that AC2 =AB2 + BC2 −2BC × BD

Ans: Proceed as sum no. 10.

12. Prove that in any triangle the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median, which bisects the third side.

acute angled triangle

acute angled

13. If A be the area of a right triangle and b one of the sides containing the right angle, prove that the length of the altitude on the hypotenuse is

 triangle

right triangle


right triangle

16. In the given figure, and E is the mid-point of CA. Prove that

mid-point

17. ABCD is a parallelogram in the given figure, AB is divided at P and CD and Q so that AP:PB=3:2 and CQ:QD=4:1. If PQ meets AC at R, prove that AR

parallelogram

parallelogram

reacangle

18. Prove that the area of a rhombus on the hypotenuse of a right-angled triangle, with one of the angles as 60o, is equal to the sum of the areas of rhombuses with one of their angles as 60o drawn on the other two sides.

the hypotenuse of a right-angled triangle

19. An aeroplane leaves an airport and flies due north at a speed of 1000 km/h. At the same time, another plane leaves the same airport and flies due west at a speed of 1200 km/h. How far apart will be the two planes after 1½ hours.

20. ABC is a right-angled isosceles triangle, right-angled at B. AP, the bisector of , intersects BC at P. Prove that

ABC is a right-angled isosceles triangle,

triangle


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