Theorem of Geometric Mean

Q. In $\Delta ABC,\angle BAC={90}^{\circ }$. From A, AD is drawn perpendicular to BC. Prove that $A{D}^2=BD\times DC$. [4 MARKS]

Q. In triangle ABC, if $AP\perp BC$ and $A{C}^2=B{C}^2-A{B}^2$, then prove that $P{A}^2=PB\times CP$.

Q.

Triangle PQR is an isosceles right triangle right angled at Q.If PR= .What is the value of each of the equal sides?

1. 

2. 

3. 5

4. 2

Q. In the given figure, B and C are the points on the opposite sides of a road and AD is a tower standing in between the road. If points A, B and C form a triangle such that AB = 6 m, $BC=4\sqrt{3}$ m and $AC=2\sqrt{3}$ m then which of the following option is correct?

दिये गये चित्र में, बिन्दु B व C एक सड़क के विपरीत ओर है तथा सड़क के मध्य एक टॉवर AD स्थित है। यदि बिन्दु A, B व C एक त्रिभुज निर्मित करते हैं जबकि AB = 6 m, $BC=4\sqrt{3}$ m तथा $AC=2\sqrt{3}$ m हैं, तब निम्नलिखित में से कौनसा विकल्प सही है?

1. Height of the tower (AD) is 3 m
   
   टॉवर (AD) की ऊँचाई 3 m है
2. ∠BAC = 60°
3. $AD=2\sqrt{3}$ m
4. $BD=\sqrt{3}$ m

Q.

Given below are the measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, using the RHS congruence rule. In the case of congruent triangles, write the result in symbolic form.

$$\begin{gathered} △ABC△PQR \\ \angle B=90°,AC=8cm,AB=3cm\angle P=90°,PR=3cm,QR=8cm \end{gathered}$$

Q. 11, 60 ___ form a right-angled triangle.
The answer should be an integer.

Q.

If sides of a triangle are proportional then angles are equal??

Q. Two right triangles $ABC$ and $DBC$ are drawn on the same hypotenuse $BC$ and on the same sides of $BC$. If $AC$ and $DB$ intersect at $P$, prove that $AP\times PC=BP\times PD$.

Q.

In an equilateral triangle PQR, if p, q and r denote the lengths perpendiculars from P, Q, R respectively on the opposite sides, then

1. p ≠ q ≠ r

2. p = q = r

3. p ≠ q = r

4. p = q ≠ r

Q. Triangle $ABC$ is right angled at $C$ and $CD$ is perpendicular to $AB$. Prove that ${BC}^2\times AD={AC}^2\times BD$.

Q. For a right-angled triangle, two small sides are of $6$cm and $8$cm length. Length of third side will be

1. $9$cm
2. none of these
3. $10$cm
4. $14$cm

Q. In a right triangle PQR and DEF, if $hypotenusePQ=hypotenuseEF$ and side $PR=DE$ then $△PQR\cong$ $△.....$

1. $EFD$
2. $EDF$
3. $DEF$
4. $FED$

Q.

Q. In the following figure angle $PQR={90}^{\circ }$ and $\overline{SQ}$ is perpendicular to PR. Show that $Q{R}^2\times P{S}^2=P{Q}^2\times Q{R}^2$

Q. $L$ and $M$ are the mid-point of $AB$ and $BC$ respectively of $△ABC$ right-angled at $B$. Prove that $4L{C}^2=A{B}^2+4B{C}^2$.

Q. In a right triangle with sides $a$ and $b$ and hypotenuse $C$ the altitude drawn on the hypotenuse is $x$. Then which one of the following is correct ?

1. $ab=x^2$
2. $\frac{1}{a}+\frac{1}{b}=\frac{1}{x}$
3. $a^2+b^2=2x^2$
4. $\frac{1}{x^2}=\frac{1}{a^2}+\frac{1}{b^2}$

Q.

$△ABC$ and $△ADC$ are two right triangles with common hypotenuse $AC$. Prove that $\angle CAD=\angle CBD$.

Q. The sides of a right-angled triangle are $21$ and $28$ feet; find the length of the perpendicular drawn to the hypothenuse from the right angle.

Q. In $\Delta ABC$, $\angle A$ is a right angle. If $\overline{AD}\perp \overline{BC}$ and $D\epsilon \overline{BC}$, then which of the following statements are true?

1. $A{D}^2=BD.DC$
2. $B{C}^2=AD.DC$
3. $A{B}^2=BD.BC$
4. $A{C}^2=CD.DB$

Q. If the triangle formed by $A(8,-10),B(7,-3)$ and $C(0,p)$ is right angled at $B$, find the value of $p$.

Q. State true or false:

1. true
2. false

Q. Which of the following can form a right angled triangle ?

1. $14,15,26$
2. $9,10,13$
3. $12,16,20$
4. None of these

Q.

In an equilateral triangle PQR, if p, q and r denote the lengths perpendiculars from P, Q, R respectively on the opposite sides, then –

1. p ≠ q ≠ r

2. p = q = r

3. p ≠ q = r

4. p = q ≠ r

Q.

Which of the following numbers can form sides of a right angled triangle?

1. 10 cm , 6 cm , 8 cm

2. 2 cm , 17 cm , 9 cm

3. 4 cm , 5 cm , 9 cm

4. 13 cm , 27 cm , 15 cm

Q. In $△PQR,m\angle Q={90}^o$ and $\overline{QM}$ is an altitiude; $M\in \overline{PR}$. If $QM=12,PR=26$; then find $PM$ and $RM$; If $PM<RM$; then find $PQ$ and $QR$.

Q.

In an equilateral triangle PQR, if p, q and r denote the lengths perpendiculars from P, Q, R respectively on the opposite sides, then –

1. p ≠ q ≠ r

2. p = q = r

3. p ≠ q = r

4. p = q ≠ r

Q.

7.(b) 7. in fig. 12.10 ,

$$\begin{gathered} \angle 1\cong \angle 2and\angle 5\cong \angle 6. \\ \left(b\right)is\angle 7\cong \angle 8? \\ givereasoninsupportofyouranswer. \end{gathered}$$

Q. Sides of triangle are given below. Determine which of them are right triangles. In case of a right triangle, write the length of its hypotenuse.

1. 7 cm, 24 cm, 25 cmj
2. 3 cm, 8 cm, 6 cm
3. 50 cm, 80 cm, 100 cm
4. 13 cm, 12 cm, 5 cm

Q. In figure, ABC is a right triangle right angled at A. D lies on BA produced and DE$\perp$BC, intersecting AC at F. If $\angle AFE={130}^o$, find $\angle ABC$.

Q. The base and hypotenuse of a right triangle are respectively $5cm$ and $13cm$ long. Its area is

1. $25{cm}^2$
2. $28{cm}^2$
3. $30{cm}^2$
4. $40{cm}^2$