Thales Theorem

Q.

In a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides. Prove it

Q. Prove "If a line 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".

Q.

If a line intersects sides AB and AC of a $\Delta ABC$ at D and E respectively and is parallel to BC, prove that $\frac{AD}{AB}=\frac{AE}{AC}$. [2 MARKS]

Q. In $\Delta$ ABC, AC = 15 cm and $DE\parallel BC$. If $\frac{AB}{AD}=3$, find EC.

1. 5 cm
2. 10 cm
3. 2.5 cm
4. 9 cm

Q.

The area bounded between the parabola $y^2=4x$ and the line$y=2x-4$ is equal to

1. $\frac{17}{3}$ square units

2. $\frac{19}{3}$ square units

3. $9$ square units

4. $15$ square units

Q. In $\Delta$ ABC, if DE $\parallel$ BC, then which of the following is stated by Basic Proportionality Theorem?

1. $\frac{AB}{DB}=\frac{EC}{AC}$
2. $\frac{AD}{DB}=\frac{AE}{EC}$
3. $\frac{AB}{DB}=\frac{AE}{EC}$
4. $\frac{AD}{DB}=\frac{AC}{EC}$

Q. In a triangle ABC AD is perpendicular to BC such that AB squared is equal to BD into CD prove that ABC is a right triangle right angle at A

Q. In the figure, PQ || AB, CQ = 4.8 cm, QB = 3.6 cm and AB = 6.3 cm. If AP = x, length of AC in terms of x is .

1. $\frac{8}{3}x$
2. $\frac{5}{3}x$
3. $\frac{4}{3}x$
4. $\frac{7}{3}x$

Q.

The sides $AB,BC\&CA$ of a triangle $ABC$ have $3,4\&5$ interior points respectively on them.

Find the number of triangles that can be constructed using these interior points as vertices.

1. $144$

2. $225$

3. $205$

4. None of these

Q. ABCD is a parallelogram with diagonal AC. If a line XY is drawn such that it cuts the side AD at point Z and XY $\parallel$ AB, then $\frac{BX}{XC}=$

1. $\frac{AY}{AC}$
2. $\frac{DZ}{AZ}$
3. $\frac{AZ}{ZD}$
4. $\frac{AY}{YC}$

Q.

In the given figure what is the relation between PA and PB.

1. PA > PB

2. PA <PB

3. PA = PB

4. PAPB

Q.

The area enclosed between the curves $y=x^3$ and $y=\sqrt{x}$ is

1. $\frac{5}{3}$

2. $\frac{5}{4}$

3. $\frac{5}{12}$

4. $\frac{12}{5}$

Q.

If $\overrightarrow{a}=2\hat{i}-3\hat{j}+4\hat{k}$, find its components in $yz$ plane

Q. In $\Delta ABC$, AC = 15 cm and DE || BC. If $\frac{AB}{AD}=3$, find EC.

1. 9 cm
2. 10 cm
3. 5 cm
4. 2.5 cm

Q.

The line segment parallel to any sides of a triangle divides other two sides of the triangle ___ .

Q. Question 7
Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side.

Q. What value of $x$ will make $DE\parallel AB$, in the given figure?
$$AD=8x+9,CD=x+3,BE=3x+4,CE=x.$$

Q. In $\Delta$ ABC, a line DE is drawn joining the midpoints of AB and BC. Which of the following statements is true?

1. DE intersects AC inside $\Delta$ ABC
2. DE intersects AC outside $\Delta$ ABC
3. DE is parallel to BC
4. DE is parallel to AC

Q. In the givn figure if DE ||BC, $\frac{AB}{AD}=10$ and $AC=50$, Find EC.

1. 40
2. 45
3. 20
4. 30

Q. $Inthe\Delta ABC,DE||BCand\frac{AD}{DB}=\frac{3}{5}.$ If $AC=5.6cm$, then $AE=$____

1. 3.5 cm
2. 2.5 cm
3. 2.1 cm
4. 2 cm

Q. D is the midpoint of side BC of triangle ABC D and e is the midpoint of AD. BE produced meet AC at point F. Prove that BE:BF=3:4

Q. In triangle $ABC,D$ is a point on $AB$ and $E$ is a point on $AC$ such that $DE||BC$. If $\frac{AD}{AB}=\frac{AE}{x}$, Then $x$ is _______.

1. $AC$
2. $CD$
3. $BA$
4. $AE$

Q. If LM || AB, AL = 2x - 4, AC = 4x, BM = x - 2 and BC = 2x + 3, then x is ____.

1. $\frac{4}{5}$
2. $\frac{5}{4}$
3. 2
4. 1

Q.

In a triangle ABC, BC ||DE If AD=1.5, DB=4.5, AE=1.3, EC=?

1. 2.6

2. 3

3. 3.6

4. 3.9

Q. In $\Delta ABC$, a line DE is drawn joining the midpoints of AB and BC. Which of the following statements is true?

1. DE is parallel to BC
2. DE is parallel to AC
3. DE intersects AC outside $\Delta$ ABC
4. DE intersects AC inside $\Delta$ ABC

Q.

In the given figure, AB & DE are ⏊ to BC.

If AB = 6 cm, DE = 4 cm & AC = 15 cm.

1. 8 cm, 27 : 16

2. 15 cm, 81 : 49

3. 10 cm, 9 : 4

4. 15 cm, 81 : 4

5. 10 cm, 9 : 16

Q.

In the given figure, ABC & CEF are 2 ∆’s, BA || CE & AF: AC = 5: 8.

1. 5 cm, 25 : 16

2. 10 cm, 25 : 64

3. 15 cm, 64 : 25

4. 15 cm, 64 : 125

5. 10 cm, 9 : 16

Q.

If the radius of $x^2+y^2+z^2–4x–4y+2z+K=0$is $5\mathrm{units}$. Find $K$.

Q.

In $\Delta ABC$, a line DE is drawn joining the midpoints of AB and BC. Which of the following statements is true?

1. DE is parallel to AC

2. DE is parallel to BC

3. DE intersects AC outside $\Delta$ ABC

4. DE intersects AC inside $\Delta$ ABC

Q.

In the figure, AB || DC, ∠BCE = 80^o, & ∠BAC = 25^o. Find ∠CBD, ∠ADC.

1. 75^o, 50^o

2. 25^o, 60^o

3. 55^o, 100^o

4. 75^o, 105^o

5. 55^o, 105^o

6. 25^o, 45^o