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Class X worksheet Triangles
Mathematics
Sikkim Manipal University
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DAV PUBLIC SCHOOL, POKHARIPUT
SUBJECT-MATHEMATICS, CLASS- X
CHAPTER-6 (TRIANGLES)
WORKSHEET-BASIC
Time – ¾ hr MM-
Fill in the blanks ( 2 X 1 = 2)
- Two polygons of the same number of sides are similar, if their corresponding angles are_____ and their corresponding sides are ________.
- If ∆𝐴𝐵𝐶 ~∆𝑃𝑄𝑅,perimeter of ∆𝐴𝐵𝐶 = 32𝑐𝑚, perimeterof ∆𝑃𝑄𝑅 = 48𝑐𝑚 & 𝑃𝑅 = 6𝑐m, then the length of 𝐴𝐶 = _______. Choose the correct option ( 2 X 1 = 2)
- ∆𝐷𝐸𝐹~∆𝐴𝐵𝐶. If DE : AB =2 :3 and ar (∆𝐷𝐸𝐹) is equal to 44 sq, then ar(∆𝐴𝐵𝐶)in sq. units is a. 99 b. 120 c. 1769 d.
- A man goes 15m due east and then 20mdue north. His distance from the starting point is a. 35m b c. 25m d. 15m Answer the following questions: (2 x 1 =2)
- It is given that ∆ DEF ∆ RPQ. Is it true to say that D=R and F=P? Why?
- ABC is an isosceles triangle with AC = BC. If 𝐴𝐵 2 = 2𝐴𝐶 2. Prove that ABC is a right triangle. Short Answer Type – I (2 x 2 =4)
- Diagonals AC and BD of a trapezium ABCD with AB ∥ DC intersect each other at the point O. Using a similarity criterion for two triangles, show that 𝑂𝐴𝑂𝐶 = 𝑂𝐵𝑂𝐷.
- If the areas of two similar triangles are equal, prove that they are congruent. Short Answer Type – II ( 2 x 3 =6)
- O is any point inside a rectangle ABCD. Prove that 𝑂𝐵 2 + 𝑂𝐷 2 = 𝑂𝐴 2 + 𝑂𝐶 2. 10 the given figure, DE ∥ OQ and DF ∥ OR. Show that EF ∥ QR.
Long Answer Type -(1 x 4 =4)
11 AB, BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that ∆𝐴𝐵𝐶~∆𝑃𝑄𝑅.
DAV PUBLIC SCHOOL, POKHARIPUT
SUBJECT-MATHEMATICS, CLASS- X
CHAPTER-6 (TRIANGLES)
WORKSHEET-STANDARD
Time – ¾ hr MM-
Fill in the blanks ( 2 X 1 = 2)
- In two similar triangles ABC and DEF, AC= 3cm and DF=5cm. the ratio of the area of two triangles is ________.
- A ladder 26m long reaches a window 24m above the ground. The distance of the foot of the ladder from the base of the wall is ________. Choose the correct option ( 2 X 1 = 2)
- In ∆𝐴𝐵𝐶 𝑎𝑛𝑑 ∆ 𝐷𝐸𝐹, ∠𝐵 = ∠𝐸, 𝐵𝐴 𝐷𝐸 =
𝐵𝐶
𝐸𝐹, then
a. ∆𝐴𝐵𝐶∆𝐷𝐸𝐹 b. ∆𝐴𝐵𝐶∆𝐸𝐷𝐹 c. ∆𝐴𝐵𝐶∆𝐹𝐸𝐷 d. ∆𝐴𝐵𝐶∆𝐸𝐹𝐷
4. If ∆𝐴𝐵𝐶~∆ 𝑃𝑄𝑅, 𝑎𝑟(∆ 𝐴𝐵𝐶)𝑎𝑟(∆𝑃𝑄𝑅) = 94 , 𝐴𝐵 = 18𝑐𝑚 & 𝐵𝐶 = 15𝑐𝑚, then PR is equal to
a. 10cm b. 12cm c. 203 𝑐𝑚 d. 8 cm
Answer the following questions: (2 x 1 =2)
5. If ABC is an equilateral triangle with AD ⊥ BC, then prove that 𝐴𝐷 2 = 3𝐷𝐶 2
6. In ∆𝐴𝐵𝐶, ∠𝐴 is acute. BD and CE are ⊥s on AC and AB respectively. Prove that
𝐴𝐵 × 𝐴𝐸 = 𝐴𝐶 × 𝐴𝐷.
Short Answer Type – I ( 2 x 2 =4)
7. In a ∆𝐴𝐵𝐶, AD is a median and E is the midpoint of AD. If BE is produced it meets
AC in F. show that AF= 1/3 AC.
8. O is any point inside a rectangle ABCD. Prove that 𝑂𝐵 2 + 𝑂𝐷 2 = 𝑂𝐴 2 + 𝑂𝐶 2.
Short Answer Type – II ( 2 x 3 =6)
9. Prove that the ratio of the altitudes of two similar triangles is equal to the ratio of
their corresponding sides.
10.∆𝐴𝐵𝐶 is right angled at B. side BC is trisected at points D and E. prove that
8𝐴𝐸 2 = 3𝐴𝐶 2 + 5𝐴𝐷 2
Long Answer Type -(1 x 4 =4)
11 an equilateral triangle ABC, D is any point on side BC such that BD = 13 BC. Prove
that 9𝐴𝐷 2 = 7𝐴𝐵 2.
EXTRA QUESTIONS
In the given figure, AD ⊥ BC. Prove that 𝐴𝐵 2 + 𝐶𝐷 2 = 𝐵𝐷 2 + 𝐴𝐶 2
BL and CM are medians of a triangle ABC right angles at A.
Prove that 4(𝐵𝐿 2 + 𝐶𝑀 2 ) = 5𝐵𝐶 2
If AD ⊥ BC, and BD = 13 CD. Prove that 2𝐶𝐴 2 = 2𝐴𝐵 2 + 𝐵𝐶 2
State and prove Pythagoras theorem.
ABC is a triangle in which ∠𝐴𝐵𝐶 > 90° and AD ⊥ BC produced. Prove that 𝐴𝐶 2 = 𝐴𝐵 2 + 𝐵𝐶 2 + 2𝐵𝐶. 𝐵𝐷
ABC is an equilateral triangle. D is a point on BC such that BD = 13 BC.
Prove that 9𝐴𝐷 2 = 7𝐴𝐵 2 7. O is any point inside a rectangle ABCD. Prove that 𝑂𝐵 2 + 𝑂𝐷 2 = 𝑂𝐴 2 + 𝑂𝐶 2
- ∆𝐴𝐵𝐶 is an isosceles triangle in which AB = AC and D is a point on BC. Prove that 𝐴𝐵 2 − 𝐴𝐷 2 = 𝐵𝐷 × 𝐶𝐷
- In right-angled triangle ABC in which∠𝐶 = 90°, if D is the mid-point of BC, prove
that 𝐴𝐵 2 = 4𝐴𝐷 2 − 3𝐴𝐶 2 10 that the sum of the squares of the diagonals of a parallelogram is equal to
the sum of the squares of its sides.
11 is a triangle in which AB=AC and D is any point in BC. Prove that 𝐴𝐵 2 −
𝐴𝐷 2 = 𝐵𝐷. 𝐶𝐷 12 ∆𝐴𝐵𝐶, 𝐷𝐸 ∥ 𝐵𝐶, where D and E are points on AB and AC respectively. if AD=2cm and DB= 3cm, then find the ratio of ar(∆𝐴𝐷𝐸) to ar(∆𝐴𝐵𝐶)
13 two sides and a median bisecting the third side of a triangle are respectively
proportional to the corresponding sides and the median of another triangle, then the two triangles are similar.
14 ∆ 𝐴𝐵𝐶, ∠𝐴𝐵𝐶 = 135°, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝐴𝐶 2 = 𝐴𝐵 2 + 𝐵𝐶 2 + 4 𝑎𝑟(∆𝐴𝐵𝐶)
15 ∆ 𝐴𝐵𝐶, 𝐴𝐷 ⊥ 𝐵𝐶 𝑎𝑛𝑑 𝐵𝐶: 𝐶𝐷 = 4: 1, 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 2𝐴𝐶 2 + 𝐵𝐶 2 = 2𝐴𝐵 2 16 ∆ 𝐴𝐵𝐶, a line XY parallel to BC cuts AB at X and AC at Y. If BY bisects∠𝑋𝑌𝐶, then
prove that BC=CY
17 D and E are points on the sides AB and AC respectively of a ∆ 𝐴𝐵𝐶 such that DE∥
BC and divides ∆ 𝐴𝐵𝐶 into two parts of equal area. Prove that 𝐵𝐷𝐴𝐵 = 2−√2 2
18 ‘A’ be the area of a right triangle and ‘b’ be one of the sides containing the right
angle, then prove that the length of the altitude on the hypotenuse is 2𝐴𝑏 √𝑏 4 +4𝐴 2 19 and CM are medians of a triangle ABC right angled at A. prove that
4(𝐵𝐶 2 + 𝐶𝑀 2 ) = 5𝐵𝐶 2 20 the bisector of an angles of a triangle bisects the opposite side, prove that the
triangle is isosceles.
Class X worksheet Triangles
Course: Mathematics
University: Sikkim Manipal University
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